Write an equation (a) in standard form and (b) in slope-intercept form for each line described. through (-2, -2), parallel to y=3

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Identify the given information: the line passes through the point (-2, -2) and is parallel to the line y = 3.

Recall that the line y = 3 is a horizontal line, which means its slope is 0. Since the new line is parallel to y = 3, it also has a slope of 0.

Write the equation of the line in slope-intercept form, which is \(y = mx + b\). Since the slope \(m = 0\), the equation simplifies to \(y = b\).

Use the point (-2, -2) to find \(b\) by substituting \(x = -2\) and \(y = -2\) into the equation \(y = b\). This gives \(-2 = b\).

Write the final equations: (a) in standard form, which for a horizontal line is \(y = -2\), and (b) in slope-intercept form, which is also \(y = -2\).

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주요 개념

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

Equation of a Horizontal Line

A horizontal line has a constant y-value for all x-values, meaning its slope is zero. Its equation is written as y = k, where k is the y-coordinate of any point on the line. For example, y = 3 is a horizontal line passing through all points with y-coordinate 3.

Parallel Lines

Parallel lines have the same slope but different y-intercepts, so they never intersect. If a line is parallel to y = 3, which is horizontal, it must also be horizontal with slope zero. Therefore, the new line will have the form y = k, where k is determined by the given point.

Forms of Linear Equations

Linear equations can be expressed in different forms. The standard form is Ax + By = C, where A, B, and C are constants. The slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. Converting between these forms helps in graphing and understanding line properties.