ResourcesSAT (Digital SAT)Linear Equations & Systems

Linear Equations & Systems

How this shows up on the SAT

Linear equations and systems are the single most-tested category on the Digital SAT Math section — roughly a third of all Math questions touch this skill in some form. You will see them as clean algebra ("solve 3x − 7 = 11"), as word problems ("a phone plan charges a

5 flat fee plus $0.10 per minute"), and as systems of two equations with deliberately engineered no-solution or infinite-solution traps. The SAT loves three specific traps on this topic: (1) systems where the two equations are multiples of each other (infinite solutions), (2) systems where the slopes match but intercepts don't (no solution), and (3) questions that ask for x + y or 2y instead of just x or y, betting that you'll stop one step early. Knowing that these traps exist cuts your careless-mistake rate in half. Strategy: rewrite every equation in slope-intercept form (y = mx + b) the moment you see it. Ninety percent of the trap categories become obvious once both sides are in that form. For word problems, name your variables in writing before you form the equation.

Concept Refresher

Read the in-depth concept guides:

Mathematics

Linear Equations: Slope, Forms, Graphing & Solving

Mathematics

Systems of Equations: Substitution, Elimination & Graphing

Linear Equations & Systems — Practice Quiz

20 SAT-styled questions. Pick an answer to see the explanation immediately.

  1. 1.What is the slope of the line 4x − 2y = 8?

  2. 2.What is the y-intercept of 3x + 5y = 15?

  3. 3.If 3x + 2y = 12 and x = 2, what is y?

  4. 4.Which equation represents a horizontal line passing through (4, −3)?

  5. 5.A line passes through (−1, 2) and (3, 10). What is its equation in slope-intercept form?

  6. 6.What is the slope of a line perpendicular to y = (2/3)x + 5?

  7. 7.Write y = −3x + 5 in standard form (Ax + By = C with A ≥ 0).

  8. 8.A line has point-slope form y − 4 = 3(x − 2). What is its y-intercept?

  9. 9.For what value of k does the system 2x + 3y = 6 and 4x + ky = 10 have NO solution?

  10. 10.For what value of c do the equations 2x + 5y = 10 and 6x + 15y = c have INFINITELY MANY solutions?

  11. 11.If 5x − 2y = 11 and 3x + 2y = 13, what is x + y?

  12. 12.Solve for x: (x + 3)/2 + (x − 1)/4 = 5

  13. 13.A car rental costs a $40 flat fee plus $0.25 per mile. Which equation gives the total cost C for m miles?

  14. 14.Two phones and three cases cost $340. One phone and one case cost

    60. What is the price of one phone?

  15. 15.The table shows points on a line: (0, 5), (1, 8), (2, 11), (3, 14). What is the equation of the line?

  16. 16.Lines L₁ and L₂ are parallel. L₁ has slope −1/4. If L₂ passes through (8, 3), what is its equation?

  17. 17.If 4x + 3y = 18 and 2x − y = 4, what is the value of y?

  18. 18.Which of the following is NOT a linear equation in one variable?

  19. 19.A line passes through (2, 5) and has slope −3. What is its equation in point-slope form?

  20. 20.The graph of y = 2x − 3 is shifted up 5 units. What is the equation of the new line?

Frequently Asked Questions

What is the fastest way to solve a linear system on the SAT?
Scan for a variable with matching or opposite coefficients — elimination is usually fastest. If one equation already has a variable alone, use substitution.
How do I tell if a system has no solution without solving it?
Rewrite both equations in slope-intercept form. If the slopes match but the y-intercepts differ, the lines are parallel and there is no solution.
When a system has infinite solutions, what should I look for?
The two equations are multiples of each other — every coefficient and the constant are scaled by the same factor. For example, 2x + 4y = 6 and 3x + 6y = 9 (everything × 1.5).

More SAT Practice