Math

Systems of Equations

6 min readMedium5-question drill

Systems of equations test whether you can find values that satisfy *two* equations at once — and the SAT loves them because they show up in word problems, graphing questions, and linear-relationship puzzles.

Where this lesson fits

Before

Linear Equations→Linear Functions→

Systems of Equations

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→Quadratic Equations →Function Notation

A system of equations is two (or more) equations that must be true at the same time. Solving means finding values for the variables that satisfy every equation in the system.

\begin{cases} x + y = 10 \\ x - y = 4 \end{cases}

The solution is $x = 7$ , $y = 3$ — both equations are satisfied.

Three solving methods. Pick whichever is fastest:

1. Substitution. Solve one equation for one variable, plug into the other.

From $x + y = 10$ : $x = 10 - y$ .
Plug into $x - y = 4$ : $(10 - y) - y = 4 \Rightarrow 10 - 2y = 4 \Rightarrow y = 3$ .
Back to $x = 10 - 3 = 7$ .

2. Elimination. Add or subtract equations to cancel a variable.

Adding the two equations directly: $(x+y) + (x-y) = 10 + 4 \Rightarrow 2x = 14 \Rightarrow x = 7$ .
Then plug into either: $7 + y = 10 \Rightarrow y = 3$ .

3. Graphing / inspection. Each linear equation is a line. The solution is where the lines cross. The SAT sometimes gives you a graph and asks you to read off the intersection.

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Which method should I use?

Is one variable already isolated (e.g. $x = ...$) or easy to isolate?

Yes ↓

→ Use SUBSTITUTION — plug into the other equation

No ↓

Are the coefficients of one variable opposites or simple multiples?

Yes ↓

→ Use ELIMINATION — add or subtract to cancel that variable

No ↓

→ Use ELIMINATION but multiply one equation first to set up cancellation

Special cases:

No solution: the lines are parallel (same slope, different y-intercepts). The system never has a common point.
Infinitely many solutions: the lines are the same (same slope, same y-intercept). Every point on the line satisfies both equations.

Quick test for slopes-and-intercepts: rewrite both equations in $y = mx + b$ form. If the $m$ 's differ, there's exactly one solution. If $m$ 's match but $b$ 's don't, no solution. If both match, infinite.

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Three possible outcomes for a 2-line linear system

Geometric picture	Algebraic signal	Number of solutions
Lines cross at one point	Different slopes	Exactly 1
Lines are parallel	Same slope, different y-intercepts	0 (no solution)
Lines are identical	Same slope, same y-intercept	Infinite

Word problems: translate two facts into two equations. "Tickets cost $12 for adults and $8 for students. They sold 100 tickets for $1,000." Two equations: $a + s = 100$ and $12a + 8s = 1000$ .

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Quick check

Pick a method (substitution if a variable is already isolated; elimination if coefficients cancel nicely). Solve, then plug your values back into BOTH equations to verify.

If 2x + y = 10 and x - y = 2, what is the value of x?

Worked examples

Example 1

Solve the system:

\begin{cases} 2x + 3y = 13 \\ 4x - y = 5 \end{cases}

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Example 2

For what value of $k$ does the system have no solution?

\begin{cases} 3x + 5y = 12 \\ 6x + 10y = k \end{cases}

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Common pitfalls

Sign errors during elimination

When subtracting one equation from another, distribute the negative to EVERY term. " $(2x + y) - (x - y) = 2x + y - x + y$ " — note the $+y$ on the right, not $-y$ . Sign mistakes here turn a 30-second problem into a wrong answer.

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Forgetting to find the second variable

Once you solve for one variable, plug back in to find the other. The SAT often asks for both values, or for a combination like $x + y$ . A solution isn't complete until both are found.

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Confusing 'no solution' with 'one variable is zero'

If $x = 0$ and $y = 5$ , that IS a solution (the point $(0, 5)$ ). 'No solution' means the system has no values that satisfy both — typically when the lines are parallel.

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Translation errors in word problems

"Twice as many students as adults" means $s = 2a$ , NOT $a = 2s$ . Read the relationship in the order it's stated and assign carefully. Test with actual numbers if unsure.

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Key takeaways

A solution to a system satisfies all equations simultaneously.
Use substitution when one variable is easy to isolate; use elimination when coefficients cancel cleanly.
Two lines: cross at one point (one solution), are parallel (no solution), or are identical (infinite solutions).
If one equation is a scalar multiple of the other, the system is either infinite or no-solution — never one solution.
Always plug the answer back into both equations to verify.

Tracks your progress across lessons.

Try it yourself

5 practice questions on Systems of Equations, drawn from the question bank. The tutor is one click away if you get stuck.

Where this lesson fits

Before

Linear Equations→Linear Functions→

Systems of Equations

You are here

→Quadratic Equations →Function Notation