Solve systems using substitution and elimination. | Systems of two linear equations in two variables | SAT Math

Use substitution or elimination, whichever cancels faster.

核心知识

Use substitution when one variable is already isolated or easy to isolate; use elimination when you can add or subtract the equations to cancel a variable.

深入理解

Both methods do the same thing — reduce two equations in two unknowns down to one equation in one unknown. The choice between them is about which path is shorter.

Substitution works best when one equation already looks like $𝑦 = \dots$ or $𝑥 = \dots$ . Plug that expression into the other equation, solve, then back-substitute.

Elimination works best when a variable has the same or opposite coefficient in both equations. If needed, multiply one or both equations by a constant so the coefficients match, then add or subtract to cancel that variable.

After finding one variable, always substitute back into an original equation to find the other. A quick check — plugging both values into both equations — catches arithmetic errors before you move on.

分步讲解

Decide: Is a variable already isolated (→ substitution) or do coefficients align (→ elimination)?
Substitution path: Solve one equation for a variable, substitute into the other, solve the resulting one-variable equation.
Elimination path: Multiply equations if needed so one variable's coefficients are equal or opposite, then add/subtract.
Back-substitute to find the second variable.
Check both values in both original equations.

常见误解

Forgetting to distribute when substituting an expression — e.g., replacing $𝑦$ in $3 𝑦$ with $2 𝑥 + 1$ but writing $3 \cdot 2 𝑥 + 1$ instead of $3 (2 𝑥 + 1)$ .
Subtracting equations but making sign errors — writing $5 𝑥 - (- 2 𝑥)$ as $3 𝑥$ instead of $7 𝑥$ .
Finding $𝑥$ but forgetting to solve for $𝑦$ , then choosing an answer that only matches $𝑥$ .

题目

示例解析

What is the solution $(𝑥, 𝑦)$ to the system $3 𝑥 + 2 𝑦 = 16$ and $𝑥 - 2 𝑦 = - 8$ ?

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