Turn a Taxi Fare Into a Straight Line and Read the Answer Off It
Learn to model a real situation as a linear polynomial, plot its graph from a table of values, compare how the coefficient controls steepness, and read answers straight off a graph.
Why does a taxi fare make a straight line?
A taxi charges the moment you sit down, then for every kilometre.
Ride km and the fare is . Ride km: . Ride km: .
Each extra kilometre adds the same — and a constant change per unit is exactly what the previous part of this chapter called a linear relationship. Written as an equation:
where is the distance in kilometres and the fare in rupees. For an km ride:
Plot a few of these points on graph paper and they line up perfectly straight. The constant rate is what straightens the graph, and the two numbers in the equation both show up on the picture: the is where the line meets the vertical axis, and the is how steeply it climbs.
This page covers the second part of the CBSE Class 9 Mathematics chapter on linear polynomials — turning a situation into an equation, drawing its graph, and reading answers back off the picture.
Ride km and the fare is . Ride km: . Ride km: .
Each extra kilometre adds the same — and a constant change per unit is exactly what the previous part of this chapter called a linear relationship. Written as an equation:
where is the distance in kilometres and the fare in rupees. For an km ride:
Plot a few of these points on graph paper and they line up perfectly straight. The constant rate is what straightens the graph, and the two numbers in the equation both show up on the picture: the is where the line meets the vertical axis, and the is how steeply it climbs.
This page covers the second part of the CBSE Class 9 Mathematics chapter on linear polynomials — turning a situation into an equation, drawing its graph, and reading answers back off the picture.
Formula
How do you turn a word problem into a linear equation?
Find the fixed part and the rate per unit. The fixed part is , the rate is :
** is the value when — the amount already there before anything starts — and is the change for each one-unit increase in , negative when the quantity falls.
Worked example 1 — taxi fare.** fixed plus per km.
At : .
Worked example 2 — mobile plan. Monthly rental plus per minute of calls.
For minutes: .
Worked example 3 — a cyclist already on the road. A cyclist is km from home and rides away at km/h. Distance from home after hours:
After hours: km.
Worked example 4 — a falling quantity. A machine worth loses of value each year. Here the rate is negative:
After years: . And the value reaches when
which is the zero of the polynomial from Part 1, now carrying a meaning in the situation.
Worked example 5 — a draining tank. A tank holds litres and drains at litres per minute.
At : litres. The tank is empty when , so minutes.
** always carries the units of .** In the taxi problem is , a fare — not a distance, even though it sits beside the kilometre term. Reading in the wrong units is the commonest way to get a correct equation and a meaningless answer.
** is the value when — the amount already there before anything starts — and is the change for each one-unit increase in , negative when the quantity falls.
Worked example 1 — taxi fare.** fixed plus per km.
At : .
Worked example 2 — mobile plan. Monthly rental plus per minute of calls.
For minutes: .
Worked example 3 — a cyclist already on the road. A cyclist is km from home and rides away at km/h. Distance from home after hours:
After hours: km.
Worked example 4 — a falling quantity. A machine worth loses of value each year. Here the rate is negative:
After years: . And the value reaches when
which is the zero of the polynomial from Part 1, now carrying a meaning in the situation.
Worked example 5 — a draining tank. A tank holds litres and drains at litres per minute.
At : litres. The tank is empty when , so minutes.
** always carries the units of .** In the taxi problem is , a fare — not a distance, even though it sits beside the kilometre term. Reading in the wrong units is the commonest way to get a correct equation and a meaningless answer.
How do you draw the graph of a linear polynomial?
Make a small table of values, plot the points, and join them with a straight line drawn right across the grid rather than stopping at the end points.
**Worked example 1 — plot .**
- gives
- gives
- gives
- gives
- gives
- gives
Check one entry: at , . Correct.
The six points lie on one straight line. It crosses the y-axis at .
**Worked example 2 — plot .**
- gives
- gives
- gives
- gives
Check: at , . Correct. The line crosses the y-axis at .
Use three points, not two. Two points always lie on some straight line, so a mistake in one of them cannot be detected — the line simply comes out wrong with no warning. A third point that misses proves there is an arithmetic error, and that is the only reason to plot it. This is the single habit worth building in this chapter.
Where the line meets the axes.
- The y-axis is crossed at , since leaves only the constant
- The x-axis is crossed where , which is — the zero of the polynomial from Part 1
For the zero is , and the check confirms it: . So the line cuts the x-axis at , between and .
Choose a sensible scale. For the taxi fare, runs over a few kilometres while runs into hundreds of rupees, so the two axes need different scales — perhaps cm for km across and cm for up. Label both axes with the quantity and its unit, and state the scale used.
**Worked example 1 — plot .**
- gives
- gives
- gives
- gives
- gives
- gives
Check one entry: at , . Correct.
The six points lie on one straight line. It crosses the y-axis at .
**Worked example 2 — plot .**
- gives
- gives
- gives
- gives
Check: at , . Correct. The line crosses the y-axis at .
Use three points, not two. Two points always lie on some straight line, so a mistake in one of them cannot be detected — the line simply comes out wrong with no warning. A third point that misses proves there is an arithmetic error, and that is the only reason to plot it. This is the single habit worth building in this chapter.
Where the line meets the axes.
- The y-axis is crossed at , since leaves only the constant
- The x-axis is crossed where , which is — the zero of the polynomial from Part 1
For the zero is , and the check confirms it: . So the line cuts the x-axis at , between and .
Choose a sensible scale. For the taxi fare, runs over a few kilometres while runs into hundreds of rupees, so the two axes need different scales — perhaps cm for km across and cm for up. Label both axes with the quantity and its unit, and state the scale used.
What happens to the graph when you change the coefficient of x?
The coefficient controls the steepness, and its sign controls the direction. Compare the family — all of these pass through the origin, because .
**Worked comparison at :**
- gives
- gives
- gives
- gives
- gives
- gives
The same step across in produces a bigger step up in as grows. **So a larger means a steeper line.**
At the gap widens further: gives while gives .
What the sign does.
- : the line rises from left to right
- : the line falls from left to right
- : the graph is a horizontal line — which is why the definition of a linear polynomial insists
** and are equally steep**, one climbing and one falling, and they are mirror images of each other in the x-axis. The steepness comes from ; the direction comes from the sign. Those are two separate pieces of information carried by one number.
**And changing does something entirely different.** Compare , and . All three have the same coefficient, so all three are equally steep — the three lines are parallel. What changes is where each one meets the y-axis: at , and . Changing slides the line up or down without tilting it.
So the two numbers in do two independent jobs: ** tilts the line, slides it.** That separation is what makes it possible to read either number off a graph without knowing the other.
**Worked comparison at :**
- gives
- gives
- gives
- gives
- gives
- gives
The same step across in produces a bigger step up in as grows. **So a larger means a steeper line.**
At the gap widens further: gives while gives .
What the sign does.
- : the line rises from left to right
- : the line falls from left to right
- : the graph is a horizontal line — which is why the definition of a linear polynomial insists
** and are equally steep**, one climbing and one falling, and they are mirror images of each other in the x-axis. The steepness comes from ; the direction comes from the sign. Those are two separate pieces of information carried by one number.
**And changing does something entirely different.** Compare , and . All three have the same coefficient, so all three are equally steep — the three lines are parallel. What changes is where each one meets the y-axis: at , and . Changing slides the line up or down without tilting it.
So the two numbers in do two independent jobs: ** tilts the line, slides it.** That separation is what makes it possible to read either number off a graph without knowing the other.
How do you read answers off a graph you have drawn?
Go across from the axis you know, up or down to the line, then across to the other axis — and then check the reading in the equation.
**Worked example 1 — the taxi fare graph, .**
- *What does a km ride cost?* Go up from to the line and across: the reading is . Check: . Correct
- *How far does take you?* Go across from to the line and down: km. Check: . Correct
- What does the y-intercept mean? At the fare is — the fixed charge before any distance is covered
**Worked example 2 — the draining tank, .**
- *How much water is left after minutes?* Reading up from : litres. Check: . Correct
- When is the tank empty? The line meets the x-axis at , so after minutes
- *What does mean?* The volume at the start, in litres
The graph gives an approximate answer; the equation gives an exact one. Reading off a grid depends on how finely it is ruled and how carefully the line was drawn, so a graph reading of "about " is a sensible answer from a picture. Substituting into the equation turns it into exactly .
Use each to check the other. If the reading and the substitution disagree by more than the width of a grid square, something is wrong — most often a mis-plotted point, so go back to the table.
Two habits that prevent most lost marks. Always state the units with a reading — , not . And check a reading is sensible for the situation: the tank cannot hold litres, so a negative reading means has gone past minutes, beyond where the model applies at all.
That last point is worth pausing on. The equation keeps producing numbers for or , and all of them are negative and all of them are meaningless — the tank stopped draining when it emptied. A model applies over a limited range, and the graph makes that range visible in a way the equation does not.
**Worked example 1 — the taxi fare graph, .**
- *What does a km ride cost?* Go up from to the line and across: the reading is . Check: . Correct
- *How far does take you?* Go across from to the line and down: km. Check: . Correct
- What does the y-intercept mean? At the fare is — the fixed charge before any distance is covered
**Worked example 2 — the draining tank, .**
- *How much water is left after minutes?* Reading up from : litres. Check: . Correct
- When is the tank empty? The line meets the x-axis at , so after minutes
- *What does mean?* The volume at the start, in litres
The graph gives an approximate answer; the equation gives an exact one. Reading off a grid depends on how finely it is ruled and how carefully the line was drawn, so a graph reading of "about " is a sensible answer from a picture. Substituting into the equation turns it into exactly .
Use each to check the other. If the reading and the substitution disagree by more than the width of a grid square, something is wrong — most often a mis-plotted point, so go back to the table.
Two habits that prevent most lost marks. Always state the units with a reading — , not . And check a reading is sensible for the situation: the tank cannot hold litres, so a negative reading means has gone past minutes, beyond where the model applies at all.
That last point is worth pausing on. The equation keeps producing numbers for or , and all of them are negative and all of them are meaningless — the tank stopped draining when it emptied. A model applies over a limited range, and the graph makes that range visible in a way the equation does not.
Exam tip
Exam tip: three points, and always label the units
Read the fixed part and the rate out of the wording. The fixed part is , the rate per unit is , and . A falling quantity has a negative : losing a year is .
Plot three points, not two. Two points always lie on a line, so an error cannot show. A third point that misses proves a mistake.
Show the table of values in the answer. Examiners award marks for the table as well as the line, and it is where an error is found.
Draw the line right across the grid, past the plotted points, and use a ruler.
Label both axes with the quantity and its unit, and state the scale — especially when the two axes need different scales.
**The y-intercept is ** and the x-intercept is , which is the zero of the polynomial.
**Larger = steeper.** Sign gives the direction: rises, falls. and are equally steep.
**Changing slides the line, changing tilts it.** , and are parallel.
Always write units with a reading — , not — and ** carries the units of .
Check every graph reading by substituting** into the equation, and sanity-check it against the situation: a tank cannot hold a negative volume.
Plot three points, not two. Two points always lie on a line, so an error cannot show. A third point that misses proves a mistake.
Show the table of values in the answer. Examiners award marks for the table as well as the line, and it is where an error is found.
Draw the line right across the grid, past the plotted points, and use a ruler.
Label both axes with the quantity and its unit, and state the scale — especially when the two axes need different scales.
**The y-intercept is ** and the x-intercept is , which is the zero of the polynomial.
**Larger = steeper.** Sign gives the direction: rises, falls. and are equally steep.
**Changing slides the line, changing tilts it.** , and are parallel.
Always write units with a reading — , not — and ** carries the units of .
Check every graph reading by substituting** into the equation, and sanity-check it against the situation: a tank cannot hold a negative volume.
Did you know
Why a straight-line model stops working past a point
The draining-tank equation is honest for the first twenty minutes and nonsense afterwards. At it reports litres, which no tank has ever held.
Nothing is wrong with the algebra. The model was built from one observation — the tank loses litres a minute — and that observation held while there was water to lose. Past there is none, and the equation has no way of knowing.
The same limit sits inside every model on this page, usually less obviously. The taxi fare is fine for a city ride and says nothing useful about a km journey, where a different rate would apply. The cyclist at km/h cannot keep that rate up for eleven hours. The machine losing a year reaches a scrap value rather than passing through zero into debt.
So a straight-line model always comes with an unstated range, and part of using one well is knowing roughly where that range ends. Graphs make the range visible — the line walks off the bottom of the grid, or into a region where the quantity cannot go — which is one good reason to draw the picture even when the equation alone would answer the question.
There is a habit of thought here worth more than the marks. A model is a description of something, not the thing itself, and it is trustworthy exactly as far as the observation behind it reaches. Extending it beyond that is extrapolation, and extrapolation is where confident arithmetic produces confidently wrong answers.
The useful question to ask of any answer is therefore not only is the arithmetic right but is this value in a range where the model still means anything. For the tank, litres fails the second test while passing the first.
Nothing is wrong with the algebra. The model was built from one observation — the tank loses litres a minute — and that observation held while there was water to lose. Past there is none, and the equation has no way of knowing.
The same limit sits inside every model on this page, usually less obviously. The taxi fare is fine for a city ride and says nothing useful about a km journey, where a different rate would apply. The cyclist at km/h cannot keep that rate up for eleven hours. The machine losing a year reaches a scrap value rather than passing through zero into debt.
So a straight-line model always comes with an unstated range, and part of using one well is knowing roughly where that range ends. Graphs make the range visible — the line walks off the bottom of the grid, or into a region where the quantity cannot go — which is one good reason to draw the picture even when the equation alone would answer the question.
There is a habit of thought here worth more than the marks. A model is a description of something, not the thing itself, and it is trustworthy exactly as far as the observation behind it reaches. Extending it beyond that is extrapolation, and extrapolation is where confident arithmetic produces confidently wrong answers.
The useful question to ask of any answer is therefore not only is the arithmetic right but is this value in a range where the model still means anything. For the tank, litres fails the second test while passing the first.
Exam relevance
How do graphs of linear polynomials feed into JEE Main?
Because every graph studied for the next three years is read the same way, and the habits fixed here are the ones used on all of them.
This is the foundation for Class 10 Linear Equations in Two Variables and Coordinate Geometry, and then for Class 11 Mathematics Straight Lines and Relations and Functions, both examined in JEE Main. The line reappears in Class 11 written as — the identical object with the coefficient renamed slope — and the point made here, that controls steepness while only slides the line, becomes the parallel and perpendicular conditions of that chapter.
The intercepts are the most reused idea. Reading off the y-axis and off the x-axis is exactly the intercept form of Class 11 Straight Lines. And the observation that the x-intercept is the zero of the polynomial connects this page to the whole of polynomial algebra: in Class 10 a quadratic's two zeros are the two points where its parabola crosses the x-axis, which is the same statement one degree higher.
Modelling a situation as an equation is what the Physics papers then do constantly. A distance-time graph for uniform motion is with the coefficient as speed and the constant as starting position — the cyclist example above, in physics vocabulary. Class 11 Motion in a Straight Line assumes a student can already read a rate off the steepness of a line.
Steepness itself is the beginning of a much larger idea. The constant change per unit found here becomes the derivative in Class 12 Continuity and Differentiability, where non-straight graphs get a steepness that varies from point to point. A straight line is the case where that steepness never changes, which is why it is where the subject starts.
What the questions look like. For board work, draw the graph of a given equation from a table, read a value off a graph, find the intercepts, and frame the equation for a stated situation are all standard — with marks for the table and the labelled axes as well as the answer. For JEE Main, a graph is almost never the question; it is the fastest route to the answer. Recognising that a relation is linear, sketching it roughly, and reading off where it crosses an axis turns many problems into one line of work.
How board and competitive emphasis differ. A board paper rewards the construction — the table, the scale, the labels, the ruled line. A competitive paper rewards the reading — seeing at a glance that a larger coefficient means a steeper line, or that two equations with the same coefficient can never meet. So the board work builds the accuracy and the competitive work needs the speed, and the table-of-values habit is what makes the fast version reliable.
The trap that costs the most marks. Mixing up what and do. A question giving two equations with the same coefficient of and asking where the lines meet has the answer nowhere — they are parallel — and the student who reaches for algebra without looking at the coefficients will spend minutes deriving a contradiction.
A second trap worth naming. Dropping the units of . In the constant is a fare of , not a distance, and an answer that reports it as kilometres has understood the graph and lost the marks anyway.
This is the foundation for Class 10 Linear Equations in Two Variables and Coordinate Geometry, and then for Class 11 Mathematics Straight Lines and Relations and Functions, both examined in JEE Main. The line reappears in Class 11 written as — the identical object with the coefficient renamed slope — and the point made here, that controls steepness while only slides the line, becomes the parallel and perpendicular conditions of that chapter.
The intercepts are the most reused idea. Reading off the y-axis and off the x-axis is exactly the intercept form of Class 11 Straight Lines. And the observation that the x-intercept is the zero of the polynomial connects this page to the whole of polynomial algebra: in Class 10 a quadratic's two zeros are the two points where its parabola crosses the x-axis, which is the same statement one degree higher.
Modelling a situation as an equation is what the Physics papers then do constantly. A distance-time graph for uniform motion is with the coefficient as speed and the constant as starting position — the cyclist example above, in physics vocabulary. Class 11 Motion in a Straight Line assumes a student can already read a rate off the steepness of a line.
Steepness itself is the beginning of a much larger idea. The constant change per unit found here becomes the derivative in Class 12 Continuity and Differentiability, where non-straight graphs get a steepness that varies from point to point. A straight line is the case where that steepness never changes, which is why it is where the subject starts.
What the questions look like. For board work, draw the graph of a given equation from a table, read a value off a graph, find the intercepts, and frame the equation for a stated situation are all standard — with marks for the table and the labelled axes as well as the answer. For JEE Main, a graph is almost never the question; it is the fastest route to the answer. Recognising that a relation is linear, sketching it roughly, and reading off where it crosses an axis turns many problems into one line of work.
How board and competitive emphasis differ. A board paper rewards the construction — the table, the scale, the labels, the ruled line. A competitive paper rewards the reading — seeing at a glance that a larger coefficient means a steeper line, or that two equations with the same coefficient can never meet. So the board work builds the accuracy and the competitive work needs the speed, and the table-of-values habit is what makes the fast version reliable.
The trap that costs the most marks. Mixing up what and do. A question giving two equations with the same coefficient of and asking where the lines meet has the answer nowhere — they are parallel — and the student who reaches for algebra without looking at the coefficients will spend minutes deriving a contradiction.
A second trap worth naming. Dropping the units of . In the constant is a fare of , not a distance, and an answer that reports it as kilometres has understood the graph and lost the marks anyway.
Key takeaways
Graphs of linear polynomials: quick revision
- A situation with a fixed part and a constant rate per unit is linear: , where is the fixed part and the rate.
- ** is the value at and is the change in per one-unit increase in , negative when the quantity falls.
- Taxi**: fixed plus /km gives ; at km the fare is .
- Mobile plan: rental plus /min gives ; minutes costs .
- Cyclist km out at km/h gives ; after h the distance is km.
- Depreciation: losing /yr gives ; after years , reaching zero at years.
- Draining tank: L at L/min gives ; L left at min, empty at min.
- To graph: build a table, plot, join with a ruled line drawn right across the grid.
- gives -values for to ; gives for to .
- Plot three points, not two — two points always lie on a line, so an error cannot show.
- The line meets the **y-axis at and the x-axis at , which is the zero** of the polynomial. For that is , since .
- **Larger means steeper.** At : gives , gives , gives , gives .
- Sign gives direction: rises, falls, is horizontal — which is why .
- and are equally steep, mirror images in the x-axis.
- ** tilts the line, slides it.** , and are parallel.
- **All of passes through the origin**, since .
- To read a graph: across from the known axis, to the line, then to the other axis — and check by substituting.
- The graph is approximate, the equation exact — use each to check the other.
- Always state units; carries the units of (, not km).
- A model applies over a limited range — reports negative litres past minutes, which the tank cannot hold.
Take your own last auto or bus fare, write its equation, sketch the line and predict what a ride twice as long would cost — then check it the next time you travel.
- ** is the value at and is the change in per one-unit increase in , negative when the quantity falls.
- Taxi**: fixed plus /km gives ; at km the fare is .
- Mobile plan: rental plus /min gives ; minutes costs .
- Cyclist km out at km/h gives ; after h the distance is km.
- Depreciation: losing /yr gives ; after years , reaching zero at years.
- Draining tank: L at L/min gives ; L left at min, empty at min.
- To graph: build a table, plot, join with a ruled line drawn right across the grid.
- gives -values for to ; gives for to .
- Plot three points, not two — two points always lie on a line, so an error cannot show.
- The line meets the **y-axis at and the x-axis at , which is the zero** of the polynomial. For that is , since .
- **Larger means steeper.** At : gives , gives , gives , gives .
- Sign gives direction: rises, falls, is horizontal — which is why .
- and are equally steep, mirror images in the x-axis.
- ** tilts the line, slides it.** , and are parallel.
- **All of passes through the origin**, since .
- To read a graph: across from the known axis, to the line, then to the other axis — and check by substituting.
- The graph is approximate, the equation exact — use each to check the other.
- Always state units; carries the units of (, not km).
- A model applies over a limited range — reports negative litres past minutes, which the tank cannot hold.
Take your own last auto or bus fare, write its equation, sketch the line and predict what a ride twice as long would cost — then check it the next time you travel.