# SAT Algebra: Linear Equations, Systems and Tips | topin

SAT guidesAlgebra

Section

Math, about 35% of questions (13–15 scored)

Skills

Linear equations in one and two variables, linear functions, systems of two equations, linear inequalities

Formats

Multiple choice and typed answers, mixed in each module

Calculator

Allowed; graphing two lines is a quick check for systems

Order

Mixed with other domains, easier to harder across each module

## How it works

Algebra questions ask you to solve linear equations, build a linear model from a description, interpret the slope or intercept of a line in context, solve or reason about a system of two equations, or work with a linear inequality. Hard versions add a constant such as k and ask for the value that gives a system no solution or infinitely many solutions.

## Worked example

In the system of equations below, k is a constant. 3x − ky = 7 6x + 4y = 5 For what value of k does the system have no solution?

1. A−8
2. B−2
3. C2
4. D8
Show the answer

B: −2

A system of two linear equations has no solution when the lines are parallel: the x and y coefficients are in the same ratio, but the constants are not. The x coefficients are 3 and 6, a ratio of 1 to 2, so the y coefficients must be too: −k / 4 = 1/2, so −k = 2 and k = −2\. Check: with k = −2 the first equation is 3x + 2y = 7; doubling it gives 6x + 4y = 14, which cannot also equal 5, so there is no solution. Choosing 2 comes from dropping the minus sign in −ky.

## A method for Algebra

1. 1  
### Translate words into an equation  
Name the variable, then write each sentence as a piece of the equation. A fixed fee is the intercept; a cost per unit is the slope.
2. 2  
### Know the forms of a line  
y = mx + b shows the slope and y-intercept directly. Two points give the slope as change in y over change in x.
3. 3  
### Solve systems the quickest way  
Use elimination when coefficients line up, substitution when one variable is already alone, and graphing when both lines are easy to enter.
4. 4  
### Special cases: compare coefficients  
No solution: same slope, different intercepts. Infinitely many: the equations are multiples of each other. Exactly one: different slopes.
5. 5  
### Inequalities: watch the sign  
Flip the inequality when you multiply or divide by a negative, and check a test point in the original inequality.

## Traps to avoid

* Answering x when the question asks for something like 3x − 2 or the value of y.
* Mixing up slope and intercept when interpreting a model, such as calling the starting fee the rate.
* Dropping a negative sign when setting coefficient ratios equal.
* Forgetting to flip an inequality after dividing by a negative number.

## Algebra FAQs

How much of SAT Math is algebra?

About 35% of the scored Math questions, or 13 to 15 of the 40, are in the Algebra domain. Advanced Math, which includes quadratics and exponentials, is another 35%.

When does a system of equations have no solution?

When the two lines are parallel: the same slope with different intercepts. In standard form, the x and y coefficients are in the same ratio but the constants are not.

Should I use the graphing calculator for algebra questions?

For systems, graphing both lines and reading the intersection is a fast check. For questions with an unknown constant, algebra is usually quicker and more reliable.

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