Linear equations describe relationships that change at a constant rate. On the SAT, you may need to solve an equation, interpret its parts, build one from a situation, or decide how many solutions it has.
Recognizing linear equations
An equation is linear when each variable has an exponent of \(1\) and variables are not multiplied by one another. Its graph is a straight line.
- Solution to an equation
- A value that makes the equation true when it replaces the variable.
\(m\) is the slope, and \(b\) is the vertical intercept.
Solving linear equations
A reliable solving process
- Simplify each side
Distribute, remove grouping symbols, and combine like terms.
- Collect variable terms
Move variable terms to one side and constants to the other.
- Isolate the variable
Use inverse operations until the variable has a coefficient of \(1\).
- Check
Substitute the result into the original equation and confirm that both sides match.
Solve a multi-step equation
Solve \(3(x-2)+5=17\).
- Distribute
\(3x-6+5=17\)
- Combine like terms
\(3x-1=17\)
- Add to both sides
\(3x=18\)
- Divide both sides
\(x=6\)
Variables on both sides, fractions, and structure
When variable terms appear on both sides, collect them on the side that keeps the arithmetic simplest. You do not have to move variables to the left; you only need all variable terms on one side.
Solve with variables on both sides
Solve \(7x+4=3x+28\).
- Collect variable terms
Subtract \(3x\) from both sides: \(4x+4=28\).
- Collect constants
Subtract \(4\): \(4x=24\).
- Isolate the variable
Divide by \(4\): \(x=6\).
One solution, no solution, or infinitely many
How the simplified statement reveals the solution type
One solution
The equation simplifies to a true value for exactly one variable value, such as \(x=4\).
No solution
The variables cancel and leave a false statement, such as \(2=7\).
Infinitely many solutions
The variables cancel and leave an identity, such as \(5=5\).
| Form | What it emphasizes | Useful clue |
|---|---|---|
| \(y=mx+b\) | Slope and vertical intercept | Use when rate and starting value matter |
| \(Ax+By=C\) | A linear relationship in two variables | Useful for intercepts and systems |
| \(y-y_1=m(x-x_1)\) | A line through a known point | Use when given a point and slope |
Modeling with linear equations
Translate the situation before solving
- Identify the unknown quantity and assign it a variable.
- Separate the fixed starting amount from the amount that changes.
- Match units before combining quantities.
- Interpret the solution in the context of the question.
The equation represents all ordered pairs \((x,y)\) that make the relationship true.
Interpret an equation in two variables
Student tickets cost $8 and adult tickets cost $12. The equation \(8s+12a=240\) represents a $240 sale. How many adult tickets were sold if \(s=12\)?
- Substitute the known value
\(8(12)+12a=240\).
- Remove the student-ticket revenue
\(96+12a=240\), so \(12a=144\).
- Interpret the result
\(a=12\), meaning 12 adult tickets.
Check your understanding
Solve \(5x+7=2x+22\).
- \(x=3\)
- \(x=5\)
- \(x=7\)
- \(x=15\)
Show answer and explanation
Answer: \(x=5\)
Subtract \(2x\) from both sides, then subtract \(7\). This gives \(3x=15\), so \(x=5\).
Key takeaways
What to remember
- Linear equations represent constant rates of change.
- Maintain equality by applying the same operation to both sides.
- Distribute carefully and combine only like terms.
- Clear fractional coefficients by multiplying every term by a common denominator.
- Look for a direct relationship between the given and requested expressions.
- Use the final simplified statement to identify the number of solutions.
- Check both the algebra and what the answer means in context.
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.