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Equations of Straight Lines

Algebra · Equations of Straight Lines

Syllabus tag: Kenya CBC | Grade 9 Mathematics | Strand 2.0 Algebra | Sub-Strand 2.2 Equations of Straight Lines (15 Lessons)

Lesson objectives

By the end of this sub-strand, you should be able to:

  • Identify the gradient of a line in real-life situations.
  • Determine the gradient of a line from two known points.
  • Determine the equation of a straight line given two points, or one point and the gradient.
  • Express a linear equation in the form y equals mx plus c and interpret both constants.

Equation of a Straight Line

A road climbing the Rift Valley escarpment gets steeper in some places than others. A ramp at a shop entrance is gentle so a wheelchair can use it. Steepness is something we can measure, and once we can measure it we can write the line down as an equation.

This is the longest topic in Grade 9 Mathematics. Take it slowly.

a) What gradient means

Gradient is a number that tells you how steep a line is. A big gradient means a steep line. A small gradient means a gentle one.

To find it, you compare how far the line goes up with how far it goes across.

Gradient is rise over run Gradient is rise over run x y −1 1 2 3 4 5 6 7 2 4 6 0 run = 4 rise = 3 A (1, 2) B (5, 5)

Going from A to B, the line rises 3 and runs 4. So the gradient is 3 divided by 4.

A line going up from left to right has a positive gradient. A line going down from left to right has a negative gradient. A flat line has a gradient of zero.

b) Gradient from two points

You do not need to count squares. If you know two points, you subtract.

Gradient from two points Gradient from two points gradient = rise / run write down the rule gradient = (5 − 2) / (5 − 1) subtract the coordinates gradient = 3 / 4 work out top and bottom m = 0.75 as a decimal

The order matters in one way only: start with the same point on the top and the bottom. If you use B first on the top, use B first on the bottom too.

c) The equation from a point and a gradient

Once you know the gradient and one point on the line, the whole line is fixed. There is only one line that passes through that point at that steepness.

Equation from a point and a gradient Equation from a point and a gradient y − y₁ = m(x − x₁) write down the rule y − 2 = 0.75(x − 1) put in point A and m y − 2 = 0.75x − 0.75 open the bracket y = 0.75x + 1.25 add 2 to both sides

d) The equation from two points

This is the same job in two steps. First find the gradient from the two points, as in part b. Then use either point with the rule in part c.

Use the point with the simpler numbers. Both points give the same final answer, so there is no advantage in choosing the harder one.

e) The form y = mx + c

Every straight line can be written as y = mx + c. This form is useful because the two letters each tell you something you can see on the graph.

m is the gradient. It controls how steep the line is, and whether it rises or falls.

c is where the line crosses the y-axis.

What m and c tell you What m and c tell you x y −1 1 2 3 4 5 6 7 2 4 6 0 c = 1.25

The solid line has a positive m, so it rises. The dashed line has a negative m, so it falls. Read c straight off the equation: in y = 0.75x + 1.25, the value of c is 1.25.

f) The x and y intercepts

An intercept is a point where the line crosses an axis.

Where the line crosses the axes Where the line crosses the axes x y −4 −2 2 4 6 −2 2 4 6 0 y-intercept x-intercept

At the y-intercept, x is always 0. At the x-intercept, y is always 0. That single fact is all you need.

Finding the intercepts Finding the intercepts y = 0.75x + 1.25 start with the equation y = 1.25 put x = 0 for the y-intercept 0 = 0.75x + 1.25 put y = 0 for the x-intercept x = −1.67 solve for x

g) Where this is used

A matatu fare with a fixed boarding charge plus a rate per kilometre is a straight line: c is the boarding charge and m is the rate. An electricity bill with a standing charge works the same way. Surveyors use gradients to set the slope of a drainage channel so water runs off at the right speed.

Words to know

  • Gradient -- the ratio of vertical change to horizontal change between two points on a line, often called rise over run.
  • y-intercept -- the value of y at the point where a line crosses the vertical axis.
  • Linear equation -- an equation whose graph is a straight line, expressible as y equals mx plus c.
  • Rise -- the difference between the y-coordinates of two points.
  • Run -- the difference between the x-coordinates of two points, taken in the same order as the rise.

:::checkpoint Check yourself

  1. A line passes through (2, 3) and (6, 11). What is its gradient?
  2. In y = 4x − 7, what is the gradient and where does the line cross the y-axis?
  3. A line has gradient 2 and passes through (1, 5). Write its equation in the form y = mx + c.
  4. Where does the line y = 3x − 6 cross the x-axis? :::

Bridge to practice

The exercises begin with reading gradients and intercepts directly, then move to calculating gradient from coordinates, rearranging into the standard form, and deriving equations from given information. When a question gives two points, resist the temptation to skip the check; substituting the second point back into your final equation catches almost every sign slip.

Check yourselfPractise Equations of Straight Lines — 10 questions →Next in MathematicsVolume of Solids →