Solve systems involving linear and nonlinear equations (e.g., line with parabola). | Systems of equations in two variables | SAT Math

Set the two y-expressions equal, solve the quadratic, and then recover both coordinates.

核心知识

Set the two expressions equal, rearrange to standard form, and solve. Each solution for $𝑥$ gives a corresponding $𝑦$ — don't forget to find both coordinates.

深入理解

Set the y-values equal.

Set the linear and nonlinear expressions for $𝑦$ equal.
Move everything to one side so you get a quadratic in standard form.
Solve and then plug each $𝑥$ back in to get $𝑦$ .

Read the question carefully: it may want one coordinate, the sum of coordinates, or both solutions.

分步讲解

Set the two expressions for y equal to each other.
Move all terms to one side to get a quadratic in standard form $𝑎 𝑥 2 + 𝑏 𝑥 + 𝑐 = 0$ .
Solve by factoring or using the quadratic formula.
Substitute each $𝑥$ -value back into either original equation to find $𝑦$ .
Answer the specific question asked (a coordinate, a sum, etc.).

常见误解

Solving for x but forgetting to find the corresponding y-values.
Sign errors when moving terms across the equals sign — double-check every term.
Using the quadratic formula but miscalculating the discriminant.

题目

示例解析

$𝑦 = 3 𝑥 - 5$
$𝑦 = 𝑥 2 - 2 𝑥 + 1$

The system of equations above has solutions $(𝑥 1, 𝑦 1)$ and $(𝑥 2, 𝑦 2)$ . What is the value of $𝑦 1 + 𝑦 2$ ?

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