SAT Math formula sheet · all 24 · Algebra · #3 of 24

No solution and infinitely many solutions on the SAT: How It Is Tested

Linear equations in one variable · Systems of two linear equations

No Solution and Infinite Solutions diagram for the SAT: No solution means parallel lines: same slope, different y-intercept. Infinitely many solutions means the same line: same
What it is really testingNo solution means parallel lines: same slope, different y-intercept. Infinitely many solutions means the same line: same slope and the same y-intercept. Get both sides into slope-intercept form, then match the slopes (and, for infinite, the intercepts) and solve for the constant.

The question from the sheet

Hard
SAT Algebraanswer 13/11
In the given equation, kk is a constant. The equation has no solution. What is the value of kk?
4(kx+7)=5211x+304(kx + 7) = \frac{52}{11}x + 30
Student-produced response: 13/11

How the diagram applies

  1. Distribute: 4kx+28=5211x+304kx + 28 = \tfrac{52}{11}x + 30.
  2. No solution means the two sides are parallel lines: same slope, different intercept. The intercepts 2828 and 3030 already differ, so set the slopes equal.
  3. 4k=52114k = \tfrac{52}{11}, so k=1311k = \tfrac{13}{11}.

Answer: 13/11

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Common Questions

It comes up as an SAT linear equations in one variable question. No solution means parallel lines: same slope, different y-intercept. Infinitely many solutions means the same line: same slope and the same y-intercept. Get both sides into slope-intercept form, then match the slopes (and, for infinite, the intercepts) and solve for the constant.

Distribute: 4kx + 28 = 5211x + 30. No solution means the two sides are parallel lines: same slope, different intercept. The intercepts 28 and 30 already differ, so set the slopes equal. 4k = 5211, so k = 1311.

No. The SAT reference sheet is twelve geometry formulas and three notes, and No Solution and Infinite Solutions is not one of them. It is one of the 24 things the test expects you to know without being given them. See what the reference sheet does cover at /blog/sat-math-reference-sheet.