PSAT 8/9 Math · Lesson 1 of 4
Linear equations and functions
Learn how to translate a relationship into an equation, keep both sides balanced, and read a slope as a rate of change.
What a linear relationship describes
A linear equation describes two quantities that change at a constant rate. If notebooks cost $3 each and a folder costs $5, the total is 3n + 5. The 3 is the rate per notebook. The 5 is paid once, so it is not multiplied by n.
A function gives one output for each allowed input. In a table, check equal steps in x and ask whether the corresponding steps in y are constant. A constant change of 4 in y for every increase of 1 in x means the slope is 4.
Solving without losing the meaning
To solve an equation, do the same operation to both sides. Subtract a constant before you divide by a coefficient. If the unknown is on both sides, gather the variable terms first. Then substitute your result into the original equation.
When the question gives two variables, two pieces of information are usually required. “Twice as many tickets as adults” is one equation. A total of 30 people is another. Write both before you start eliminating.
Worked example
A plumber charges $40 plus $25 per hour. The bill is $140. How many hours is that?
- Let h be the number of hours. The bill is 25h + 40.
- Set 25h + 40 = 140, so 25h = 100.
- h = 4. Check: 25 × 4 + 40 = 140.
Why this works. The trip fee is added once. Dividing 140 by 25 ignores that fee and gives a different answer.
Check your understanding
- Identify which number is a rate and which number is a one-time amount.
- Solve a one-variable equation and check it in the original statement.
- Read a slope in words, including units, before you graph it.