Use substitution to reduce a system to one equation in one variable. | Systems of equations in two variables | SAT Math

Pick the easiest variable to isolate, substitute, and solve the resulting one-variable equation.

核心知识

Pick the equation where isolating a variable takes the least work, solve for that variable, and plug the expression into the other equation. Now you have one equation in one unknown.

深入理解

Substitution is the primary method for SAT systems involving a quadratic. The idea: if one equation tells you what $𝑦$ equals in terms of $𝑥$ , replace every $𝑦$ in the other equation with that expression.

Choose the equation that's already solved for a variable, or the one where isolating a variable is simplest. The linear equation is almost always the better choice.

After substituting, you'll have a single equation in one variable. Solve it, then back-substitute into the simpler original equation to find the other variable. Use the simpler equation for back-substitution to reduce arithmetic errors.

分步讲解

Identify which equation has a variable that's easy to isolate.
Solve that equation for the chosen variable.
Replace that variable in the other equation with the expression you found.
Simplify and solve the resulting single-variable equation.
Plug each solution back into the simpler equation to find the other variable.

常见误解

Substituting back into the same equation you derived from — this creates a circular identity. Use the other equation.
Dropping terms or flipping signs during substitution, especially when distributing negatives.
Forgetting to distribute when substituting an expression like $10 - 3 𝑥$ into a product.

题目

示例解析

$3 𝑥 + 𝑦 = 10$
$𝑥 𝑦 = - 8$

If $(𝑥, 𝑦)$ is a solution to the system above and $𝑥 > 0$ , what is the value of $𝑦$ ?

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