What the Graph of a Linear Equation Tells You

A brief note on What the Graph of a Linear Equation Tells You. Read it before a test. Try each question at the end before you open its answer.

Equations and solutions

• A linear equation in two variables can be written as $ax + by + c = 0$, with $a$ and $b$ not both zero.
• A solution is a pair $(x, y)$ that makes it true. There are infinitely many: choose $x$, and $y$ follows.
• To test a pair, put it in the equation and compare the two sides.

The graph

• Every solution is a point, and all of them lie on one straight line.
• x-intercept: put $y = 0$. y-intercept: put $x = 0$. Two points draw the line; a third is a check.

Special lines

A rule such as a fare, $y = 2x + 3$ dollars, is a line: the slope is the charge for each kilometre and $c$ is the fixed charge.

A worked example

Draw the graph of $3x + 5y = 45$.

1. x-axis. $y = 0$: $3x = 45$, so the point is $(15, 0)$.
2. y-axis. $x = 0$: $5y = 45$, so the point is $(0, 9)$.
3. Check. $x = 5$: $15 + 5y = 45$, so $y = 6$. The point $(5, 6)$ lies on the line through the first two.

Practice with new numbers

A program to try

The program works out the worked example.

Mistakes to avoid

• Thinking a linear equation in two variables has one solution. It has infinitely many.
• Swapping the order of a pair. $(3, 4)$ means $x = 3$ and $y = 4$.
• Using the number in front of $x$ as the x-intercept. The intercept of $4x + 5y = 20$ is $5$, not $4$.
• Mixing up $x = 3$ and $y = 3$. The first runs up and down; the second runs across.

Check yourself

1. Is $(6, -2)$ a solution of $3x + 4y = 10$?

2. Where does $6x + 5y = 30$ meet the x-axis?

3. Which line is parallel to the y-axis, $x = 7$ or $y = 7$?

4. A cab charges $3$ dollars plus $2$ dollars for each kilometre. What is the fare for $9\ \text{km}$?