Question

GRAPHING Write the equation in slope-intercept form, and then graph the equation. Label the \(x\) - and \(y\) -intercepts on the graph. $$-x+y-7=0$$

Short answer

The slope-intercept form of the equation is \(y = x + 7\). The x-intercept is (-7,0) and the y-intercept is (0,7). The graph of the equation is a straight line going through these points.

Step-by-step solution

Convert equation to slope-intercept form

Slope-intercept form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is \(y\)-intercept. We need to rearrange the given equation \(-x + y - 7 = 0\) to this form. Let's add \(x\) to both sides of the equation to get \(y = x + 7\).

Identify Slope and Y-intercept

From the equation \(y = x + 7\), the slope \(m = 1\) because it's the coefficient of \(x\), and the y-intercept is \(b = 7\) because it's the constant term.

Plot the Y-intercept

Plot the y-intercept using the y-intercept value, here it's \(b = 7\). So, place a dot at the point where \(y = 7\).

Use the Slope to Find Another Point

The slope is the rise over run ratio, which is 1 in this case. From the y-intercept, 'rise' one space upward (increase \(y\) by 1) and 'run' one space to the right(increase \(x\) by 1) to plot a second point.

Draw the Line

Draw a straight line through the two points plotted. This line represents the set of all solutions to the equation.

Find and Label the X-intercept

The x-intercept is found by setting \(y = 0\) in slope-intercept equation. For \(y = x + 7\), we get \(x = -7\). This is where our line crosses the x-axis. Hence, label this point on the graph (-7,0).

Label the Y-intercept

Label the point where the line crosses the y-axis. This is our y-intercept, it is (0,7).