Guides / Skip solving systems of equations
When to skip solving a system of equations on the SAT
For questions about how many solutions a system has, compare coefficient ratios instead of solving for x and y.
If a digital SAT question about a system of two linear equations asks how many solutions exist, or asks for a constant that produces no solution or infinite solutions, you never need to solve for x and y. Compare the ratios of the coefficients instead.
The shortcut
For a system of two equations in the form ax + by = c and dx + ey = f, look at the ratios a/d, b/e, and c/f.
If a/d equals b/e but does not equal c/f, the lines are parallel: no solution. If a/d, b/e, and c/f are all equal, the two equations describe the same line: infinite solutions. If a/d does not equal b/e, the lines cross exactly once: exactly one solution. None of these require isolating x or y.
3x + 6y = 12 kx + 12y = 24
For what value of k does the system of equations above have infinitely many solutions?
Infinite solutions means every ratio must match. The y-coefficients give 6/12 = 1/2, and the constants give 12/24 = 1/2, both already equal. Setting the x-coefficient ratio to match: 3/k = 1/2, so k = 6. No need to solve the system itself, just match the ratio the other two pairs already set.
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