Use a system of equations to solve each problem. See Example 8. Find an equation of the line y = ax + b that passes through the points (-2, 1) and (-1, -2).

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Identify the general form of the equation of the line: \(y = ax + b\), where \(a\) is the slope and \(b\) is the y-intercept.

Use the two given points \((-2, 1)\) and \((-1, -2)\) to create two equations by substituting \(x\) and \(y\) values into \(y = ax + b\):

\(1 = a(-2) + b\) and \(-2 = a(-1) + b\).

Rewrite the system of equations as: \[ \begin{cases} -2a + b = 1 \\ -a + b = -2 \end{cases} \]

Solve this system of linear equations to find the values of \(a\) and \(b\), which will give you the equation of the line.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

System of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. Solving the system means finding values for the variables that satisfy all equations simultaneously. In this problem, the variables are the coefficients a and b in the line equation y = ax + b.

Slope-Intercept Form of a Line

The slope-intercept form y = ax + b expresses a line where 'a' is the slope and 'b' is the y-intercept. The slope measures the rate of change of y with respect to x, and the y-intercept is the point where the line crosses the y-axis. Finding a and b defines the line passing through given points.

Using Points to Form Equations

Each point (x, y) on a line satisfies the line's equation. Substituting the coordinates of given points into y = ax + b creates equations involving a and b. Solving these equations simultaneously yields the values of a and b that define the line passing through both points.