Write an equation for each line described. Give answers in standard form for Exercises 11–20 and in slope-intercept form (if possible) for Exercises 21–32. m=5, b=15

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Identify the given slope and y-intercept from the problem: slope \(m = 5\) and y-intercept \(b = 15\).

Recall the slope-intercept form of a line, which is given by the equation \(y = mx + b\).

Substitute the given values of \(m\) and \(b\) into the slope-intercept form to write the equation: \(y = 5x + 15\).

To write the equation in standard form, recall that the standard form of a line is \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers and \(A \geq 0\).

Rearrange the slope-intercept form \(y = 5x + 15\) by moving all terms to one side to get \(-5x + y = 15\), then multiply both sides by \(-1\) to make \(A\) positive, resulting in \(5x - y = -15\).

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주요 개념

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Slope-Intercept Form of a Line

The slope-intercept form is expressed as y = mx + b, where m represents the slope and b is the y-intercept. This form clearly shows the rate of change and the point where the line crosses the y-axis, making it easy to graph and understand linear relationships.

Standard Form of a Line

Standard form of a linear equation is written as Ax + By = C, where A, B, and C are integers, and A should be non-negative. This form is useful for analyzing and solving systems of equations and is often required for final answers in algebra problems.

Interpreting Slope and Y-Intercept

The slope (m) indicates the steepness and direction of a line, showing how much y changes for a unit change in x. The y-intercept (b) is the point where the line crosses the y-axis (x=0). Understanding these helps in writing and graphing the equation of a line from given values.