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Math foundations

SAT systems of equations: where two lines meet

A system’s solution is the point both equations agree on. See it on a graph, choose substitution or elimination with a simple decision flow, and spot no-solution and infinite-solution systems instantly.

SPSAT Practice Editorial8 min read

The short version

What you'll take away

  • The solution to a system of two linear equations is the point where their graphs intersect.
  • Use substitution when a variable is already isolated; use elimination when coefficients line up.
  • Same slope with different intercepts means no solution; equivalent equations mean infinitely many solutions.
In this article

A solution is a point both equations agree on

A system of equations is two or more equations that must be true at the same time. For two linear equations in x and y, each equation is a line, and the solution is the point that lies on both lines.
Take \(\displaystyle y = 2x + 1\) and \(\displaystyle y = -x + 7\). The first line climbs 2 for every step right; the second falls 1. They cross exactly once, at (2, 5). Check it: \(\displaystyle 2(2) + 1 = 5\) and \(\displaystyle -2 + 7 = 5\).

The study lab

Two lines, one intersection

y = 2x + 1 and y = −x + 7 cross at the system’s solution.

What to notice: The intersection is the only point that makes both equations true. Every other point on either line satisfies just one equation.

Choose your method with one quick look

Every method finds the same point. The skill is choosing the shortest route from the form the equations are already in.

See the skill

Which method should I use?

  1. Is one variable already alone?

    Look for y = … or x = … in either equation.

    Yes → substitution

    Replace that variable in the other equation. Example: 2x + 1 = −x + 7, so x = 2.

    No → keep looking

    Check whether the coefficients line up.

  2. Do the coefficients match or nearly match?

    Same or opposite coefficients on x or y, or ones that match after multiplying by a small number.

    Yes → elimination

    Add or subtract the equations to cancel one variable.

    Messy numbers

    Graph both equations in the built-in Desmos calculator and click the intersection.

  3. Check the answer in both equations

    A point that satisfies only one equation is not a solution. A quick substitution catches arithmetic and sign errors.

The digital SAT allows a calculator, including the built-in Desmos graphing calculator, on every Math question.

Elimination in a word problem

A school sells 40 tickets for a play. Adult tickets cost $12, student tickets cost $8, and sales total $400. Let a be the number of adult tickets and s the number of student tickets.
The count gives \(\displaystyle a + s = 40\), and the money gives \(\displaystyle 12a + 8s = 400\). Multiply the first equation by 8 to get \(\displaystyle 8a + 8s = 320\), then subtract it from the second: \(\displaystyle 4a = 80\), so \(\displaystyle a = 20\) and \(\displaystyle s = 20\). Check: \(\displaystyle 12(20) + 8(20) = 400\).

No solution or infinitely many solutions

Two lines do not always cross exactly once. Parallel lines have the same slope and different y-intercepts, so they never meet: the system has no solution. If the two equations describe the same line, every point on it works: infinitely many solutions.
On the SAT, these questions often hide a constant like k. Rewrite both equations in the same form and compare. For infinitely many solutions, one equation must be a multiple of the other, including the constant. For no solution, the x- and y-coefficients are proportional but the constants are not.

The study lab

Parallel lines never meet

y = 2x + 1 and y = 2x − 3 have the same slope and different intercepts.

What to notice: The lines stay 4 units apart vertically at every x, so no point satisfies both equations. This system has no solution.

Your turn

Quick practice

For what value of k does the system 3x + ky = 6 and 6x + 4y = 12 have infinitely many solutions?

Hint: What number turns the first equation into the second?

Choose one answer

Practice: try these before checking the answers

For each system, decide on a method before solving.
  • 1. Solve \(\displaystyle y = 3x - 2\) and \(\displaystyle y = x + 4\).
  • 2. Solve \(\displaystyle x + y = 10\) and \(\displaystyle x - y = 4\).
  • 3. How many solutions does \(\displaystyle y = -4x + 1\) and \(\displaystyle 8x + 2y = 6\) have?
  • 4. Pens cost $2 and notebooks cost $5. Maya buys 9 items for $27. How many notebooks did she buy?

Sources & further reading

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