For each line, (a) find the slope and (b) sketch the graph. 5x - 2y = 10

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Rewrite the given equation $5x - 2y = 10$ in slope-intercept form $y = mx + b$ by isolating $y$ on one side.

Subtract \$5x$ from both sides to get $-2y = -5x + 10$.

Divide every term by $-2$ to solve for $y$: $y = \frac{-5x}{-2} + \frac{10}{-2}$.

Simplify the fractions to get $y = \frac{5}{2}x - 5$, where the slope $m = \frac{5}{2}$ and the y-intercept $b = -5$.

To sketch the graph, plot the y-intercept $(0, -5)$ on the coordinate plane, then use the slope $\frac{5}{2}$ to rise 5 units and run 2 units to find another point, and draw the line through these points.

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주요 개념

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Slope of a Line

The slope 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). For a line in standard form Ax + By = C, the slope is found by rearranging to slope-intercept form y = mx + b, where m is the slope.

Converting Standard Form to Slope-Intercept Form

To find the slope easily, rewrite the equation from standard form (Ax + By = C) to slope-intercept form (y = mx + b) by isolating y. This process involves solving for y, which reveals the slope (m) and y-intercept (b) directly.

Graphing a Line Using Slope and Intercept

Graphing a line involves plotting the y-intercept on the coordinate plane and using the slope to find other points by moving vertically and horizontally. This method provides a visual representation of the line’s behavior and direction.