For each line, (a) find the slope and (b) sketch the graph. 4x + 3y = 12

검증된 단계별 안내

Rewrite the given equation \(4x + 3y = 12\) in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. To do this, solve for \(y\): subtract \(4x\) from both sides to get \(3y = -4x + 12\).

Divide every term by 3 to isolate \(y\): \(y = \frac{-4}{3}x + 4\). Now the equation is in the form \(y = mx + b\), where the slope \(m = -\frac{4}{3}\) and the y-intercept \(b = 4\).

Identify the slope \(m = -\frac{4}{3}\). This means for every 3 units you move to the right along the x-axis, the line goes down 4 units along the y-axis.

To sketch the graph, start by plotting the y-intercept point \((0, 4)\) on the coordinate plane.

From the y-intercept, use the slope to find another point: move 3 units to the right (positive x-direction) and 4 units down (negative y-direction), then plot this second point. Draw a straight line through these two points to complete the graph.

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주요 개념

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Slope of a Line

The slope measures the steepness of a line and is 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 Linear Equation

Graphing involves plotting points that satisfy the equation and drawing a line through them. Using the slope and y-intercept, you can start at the y-intercept on the graph and use the slope to find another point, then draw the line through these points.