The graph of a linear equation

Solutions are points

A solution $(x, y)$ is a pair of numbers. We can show a pair of numbers as a point on a graph. The number $x$ tells how far to go to the right, and $y$ tells how far to go up.

Take the equation $2x + y = 10$ from the last lesson. Its solutions were $(0, 10)$, $(1, 8)$, $(2, 6)$, $(3, 4)$, $(4, 2)$ and $(5, 0)$. Let us put these points on a graph.

Look at the points. They lie on a straight line. This is true for every linear equation in two variables.

• Every solution of the equation is a point on the line.
• Every point on the line is a solution of the equation.

So the line is a picture of all the solutions together. This is why we say a "linear" equation: its graph is a line.

How many points do we need?

Two points are enough to draw a straight line. We draw the line through them. A third point is a good check. If the third point is not on the line, we made a mistake.

The intercepts

Two points are easy to find. They are the points where the line meets the axes.

• To find where the line meets the x-axis, we put $y = 0$. This gives the x-intercept.
• To find where it meets the y-axis, we put $x = 0$. This gives the y-intercept.

Example. Draw the graph of $2x + 3y = 12$.

Step 1. Put $y = 0$: $2x = 12$, so $x = 6$. The point is $(6, 0)$.

Step 2. Put $x = 0$: $3y = 12$, so $y = 4$. The point is $(0, 4)$.

Step 3. Find a third point as a check. Put $x = 3$: $6 + 3y = 12$, so $y = 2$. The point is $(3, 2)$.

Step 4. Draw the line through $(6, 0)$ and $(0, 4)$. The point $(3, 2)$ is on it. So we are right.

Make $a$ bigger and watch the red dot move towards the origin. The number in front of $x$ changes only where the line meets the x-axis.

Using the graph

The graph helps us to find more solutions. Choose a value of $x$ on the x-axis, go up to the line, and read $y$. For the line $2x + 3y = 12$, at $x = 3$ the line is at $y = 2$.

A common mistake

Do not join the points with a curve, and do not stop the line at the last point. The line goes on in both directions. Every point on it is a solution.

Practise with new numbers

Try it in Python

This program finds the two intercepts of the line $2x + 3y = 12$.

Remember these

• The graph of a linear equation in two variables is a straight line.
• Every point on the line is a solution, and every solution is a point on the line.
• Put $y = 0$ to find the x-intercept. Put $x = 0$ to find the y-intercept.
• Two points draw the line. A third point is a check.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Where does the line $x + y = 5$ meet the x-axis and the y-axis?

2. Is the point $(3, 3)$ on the line $x + y = 5$?

3. A student plots four points of a linear equation. Three lie on a line, but one does not. What should the student do?

4. Where does the line $4x + 5y = 40$ meet the two axes?