Problem 50 - Algebra for College Students

Question

Find the \(x\) - and \(y\) -intercepts for each line and use them to graph the line. $$x+y+7=0$$

Step-by-Step Solution

Answer

The x-intercept is (-7, 0) and the y-intercept is (0, -7).

Step 1: Find the x-intercept

To find the x-intercept, set y to 0 in the equation and solve for x.\[ x + y + 7 = 0 \]Set y to 0:\[ x + 0 + 7 = 0 \]Simplify the equation:\[ x + 7 = 0 \]Solve for x:\[ x = -7 \]So, the x-intercept is (-7, 0).

Step 2: Find the y-intercept

To find the y-intercept, set x to 0 in the equation and solve for y.\[ x + y + 7 = 0 \]Set x to 0:\[ 0 + y + 7 = 0 \]Simplify the equation:\[ y + 7 = 0 \]Solve for y:\[ y = -7 \]So, the y-intercept is (0, -7).

Step 3: Plot the intercepts on the graph

Plot the x-intercept (-7, 0) and the y-intercept (0, -7) on the coordinate plane.

Step 4: Draw the line

Draw a straight line through the points (-7, 0) and (0, -7) to graph the line.

Key Concepts

x-intercept

The x-intercept of a line is the point where the line crosses the x-axis on a coordinate plane. This occurs when the value of y is zero. To find the x-intercept, set y to 0 in the original equation and solve for x. In the equation \[x + y + 7 = 0\], setting y to 0 gives us \[x + 0 + 7 = 0\]. Simplifying this, we get x + 7 = 0, and solving for x yields \[x = -7\]. Therefore, the x-intercept is the point (-7, 0).Graphing the x-intercept helps in visualizing the position of the line relative to the x-axis.

y-intercept

The y-intercept of a line is the point where the line crosses the y-axis on a coordinate plane. This is the value of y when x equals zero. To find the y-intercept, set x to 0 in the equation and solve for y. Taking our equation \[x + y + 7 = 0\], setting x to 0 gives us \[0 + y + 7 = 0\]. Simplifying this, we get y + 7 = 0, and solving for y yields \[y = -7\]. Therefore, the y-intercept is the point (0, -7).Graphing the y-intercept is crucial for understanding the interaction of the line with the y-axis.

coordinate plane

The coordinate plane is a fundamental concept in graphing linear equations. It consists of a two-dimensional surface defined by a horizontal axis (x-axis) and a vertical axis (y-axis) that intersect at a point called the origin (0,0). Each point on the plane is defined by a pair of coordinates (x,y), representing its position relative to the origin.Understanding how to plot points on the coordinate plane is essential for graphing lines and interpreting their relationships. In our example, we plotted the x-intercept (-7, 0) and y-intercept (0, -7) on this plane. A line passing through these intercepts accurately represents the equation \[x + y + 7 = 0\]. Plotting these points and then drawing a straight line through them helps visualize the solution and better understand the equation's behavior.