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. 4x+y-6=0

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Start with the given equation: $4x + y - 6 = 0$.

Rewrite the equation to solve for $y$ in terms of $x$ to get it into slope-intercept form $y = mx + b$. To do this, isolate $y$ by subtracting \$4x$ and adding \(6$ to both sides: \)y = -4x + 6$.

Identify the slope $m$ and the y-intercept $b$ from the slope-intercept form $y = mx + b$. Here, the slope $m$ is the coefficient of $x$, which is $-4$, and the y-intercept $b$ is the constant term, which is $6$.

To graph the linear function, start by plotting the y-intercept point $(0, 6)$ on the coordinate plane.

Use the slope $m = -4$ to find another point. Since slope is rise over run, from the y-intercept move down 4 units (rise = -4) and right 1 unit (run = 1) to plot a second point. Then draw a straight line through these two points to complete the graph.

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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.