Linear equations and graphs: read a constant rate of change

1. Every point gives two pieces of information

A coordinate grid has a horizontal x-axis and a vertical y-axis. They meet at the origin, written (0, 0). An ordered pair (x, y) tells you the horizontal coordinate first and the vertical coordinate second. To plot (3, 5), move three x-units right from the origin, then five y-units up. Reversing the order gives (5, 3), usually a different point.

Positive coordinates run right and up; negative coordinates run left and down. The point (−2, 4) lies two units left and four up. Each axis needs a label and a consistent scale, but the scales need not be identical. One horizontal square could mean one minute and one vertical square ten litres. Read the labels rather than judging numerical change from the picture's angle alone.

Sources: OpenStax: Prealgebra 2e: Use the Rectangular Coordinate System ↗

2. Build a table from a rule

Consider y = 2x + 3, where x is the input and y the output. The expression 2x means two multiplied by x. Substituting x values 0, 1, 2 and 3 gives y values 3, 5, 7 and 9. The resulting points are (0, 3), (1, 5), (2, 7) and (3, 9). They line up because equal input steps produce equal output steps.

A point belongs to the graph only if its coordinates satisfy the equation. Test (4, 11): 2 × 4 + 3 = 11, so it belongs. Test (4, 10): the rule still gives 11, so it does not. Two distinct points determine a straight line; calculating a third point is a useful check against substitution or plotting mistakes.

Sources: NCERT Class VIII Mathematics Exemplar: Introduction to Graphs ↗ · OpenStax: Prealgebra 2e: Graphing Linear Equations ↗

3. Slope is change divided by change

For a nonvertical line, slope means the change in y divided by the corresponding change in x. Label two points (x₁, y₁) and (x₂, y₂); the small 1 and 2 identify the points, not powers. Then slope m = (y₂ − y₁)/(x₂ − x₁). Use the same point order in both subtractions. A zero denominator would mean both points have the same horizontal coordinate.

Between (1, 5) and (3, 9), m = (9 − 5)/(3 − 1) = 4/2 = 2. Output rises two units per input unit. A negative slope means output falls as input grows. A horizontal line has slope zero. A vertical line has undefined slope and cannot be written as y = mx + b with a finite value of m.

Sources: OpenStax: Elementary Algebra 2e: Slope-Intercept Form of an Equation of a Line ↗

4. The intercept gives a starting value

In y = mx + b, b is the y-intercept: the y-value where x = 0. Thus y = 2x + 3 starts at (0, 3). The x-intercept is where y = 0. Solving 0 = 2x + 3 gives x = −1.5, so the line meets the x-axis at (−1.5, 0). An intercept may lie outside the part of the graph relevant to an application.

A directly proportional relationship has b = 0 and passes through the origin. A straight line with b different from zero is still linear but is not directly proportional. Doubling x in y = 2x + 3 does not generally double y. Distinguishing rate from starting value is essential when comparing charges, water levels or distances.

Sources: OpenStax: Prealgebra 2e: Graphing with Intercepts ↗ · OpenStax: Elementary Algebra 2e: Slope-Intercept Form of an Equation of a Line ↗

5. Solved example: a tank drains at a constant rate

An illustrative tank contains 180 litres and loses 20 litres each minute. Let t be elapsed time in minutes and V the remaining volume in litres. The rule is V = 180 − 20t. At t = 0, 2, 4 and 6, the volumes are 180, 140, 100 and 60 litres. Plot time horizontally and volume vertically. The slope is −20 litres per minute and the vertical intercept is 180 litres.

To find the emptying time, set V = 0: 180 − 20t = 0, giving t = 9 minutes. Check 20 × 9 = 180 litres removed. The physical model applies from 0 to 9 minutes. Extending the line to t = 12 gives −60 litres, an algebraic value with no meaning as remaining water. After emptying, a different rule is needed; the tank cannot continue losing water it does not contain.

Sources: OpenStax: Prealgebra 2e: Graphing Linear Equations ↗ · OpenStax: Elementary Algebra 2e: Slope-Intercept Form of an Equation of a Line ↗

6. Solved example: recover a rule from two observations

An illustrative equipment-hire model charges ₹70 for 2 hours and ₹130 for 6 hours. Assume a fixed starting charge plus a constant hourly rate. The extra 4 hours cost ₹60, so the slope is 60/4 = ₹15 per hour. Let h be hours and C cost in rupees. Write C = 15h + b. Substituting (2, 70) gives 70 = 30 + b, so b = 40.

The rule C = 15h + 40 predicts ₹100 for 4 hours. Verify the second observation: 15 × 6 + 40 = 130. The two points determine a candidate straight line, but do not prove the provider always follows it. A minimum booking period, rounded billing hours or a daily maximum could change the rule. A fitted equation and a verified policy are different kinds of evidence.

Sources: OpenStax: Elementary Algebra 2e: Slope-Intercept Form of an Equation of a Line ↗

7. Connect points only when the situation permits it

Time and volume can take intermediate values, so a continuous segment suits the tank model. The number of complete notebooks takes whole-number values, so a purchase graph may show separate points even when a line describes the pattern. Interpolation estimates between observed inputs; extrapolation goes beyond them and depends more strongly on unchanged conditions. Clearly mark illustrative data and state the model's assumptions before treating a graph as a forecast.

Sources: OpenStax: Prealgebra 2e: Graphing Linear Equations ↗

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