In Exercises 59-66, a. Rewrite the given equation in slope-intercept form. b. Give the slope and y-intercept. c. Use the slope and y-intercept to graph the linear function. 8x – 4y – 12 =0

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Start by rewriting the given equation $8x - 4y - 12 = 0$ in slope-intercept form, which is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Isolate the $y$ term on one side. Add \$4y$ to both sides and subtract $12$ from both sides to get $8x - 12 = 4y$.

Divide every term by 4 to solve for $y$: $\frac{8x}{4} - \frac{12}{4} = y$, which simplifies to $y = 2x - 3$.

Identify the slope $m$ and y-intercept $b$ from the equation $y = 2x - 3$. Here, the slope $m$ is 2, and the y-intercept $b$ is $-3$.

To graph the function, start by plotting the y-intercept at $(0, -3)$ on the coordinate plane. Then use the slope $2$ (which means rise over run = 2/1) to find another point by moving up 2 units and right 1 unit from the y-intercept, and draw the line through these points.

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주요 개념

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Slope-Intercept Form

The slope-intercept form of a linear equation is y = mx + b, where m represents the slope and b represents the y-intercept. Rewriting an equation into this form makes it easier to identify these values and to graph the line.

Slope of a Line

The slope (m) measures the steepness and direction of a line, calculated as the ratio of the change in y to the change in x (rise over run). It indicates how much y changes for a unit change in x and is essential for understanding the line's behavior.

Graphing Using Slope and Y-Intercept

Graphing a linear function involves plotting the y-intercept (where the line crosses the y-axis) and then using the slope to find other points by moving vertically and horizontally. This method provides a straightforward way to visualize the line on the coordinate plane.