Problem 23 - Understanding Elementary Algebra with Geometry

Question

Sketch the graph of the given equation. Label the intercepts. $$x+y=-5$$

Step-by-Step Solution

Answer

The graph of \(x + y = -5\) crosses the y-axis at (0, -5) and the x-axis at (-5, 0).

Step 1: Rewrite the Equation

Rewrite the equation in slope-intercept form, which is easier to graph. The original equation is: \(x + y = -5\)Subtract \(x\) from both sides to get:\(y = -x - 5\)

Step 2: Identify the Slope and Y-intercept

From the slope-intercept form \(y = mx + b\), identify the slope \(m\) and the y-intercept \(b\). In this equation, \(m = -1\) and \(b = -5\), meaning the line has a slope of -1 and intersects the y-axis at -5.

Step 3: Find the Y-intercept

The y-intercept is the point where the graph crosses the y-axis. Set \(x = 0\) in the equation \(y = -x - 5\):\(y = -0 - 5\)\(y = -5\)So, the y-intercept is (0, -5).

Step 4: Find the X-intercept

The x-intercept is the point where the graph crosses the x-axis. Set \(y = 0\) in the equation \(y = -x - 5\):\(0 = -x - 5\)\(x = -5\)So, the x-intercept is (-5, 0).

Step 5: Plot the Intercepts

Plot the points (0, -5) and (-5, 0) on the coordinate plane. These are the y-intercept and x-intercept, respectively.

Step 6: Draw the Line

Connect the plotted points with a straight line. This line represents the graph of the equation \(x + y = -5\).

Step 7: Label the Intercepts

Label the intercept points on the graph: (0, -5) as the y-intercept and (-5, 0) as the x-intercept.

Key Concepts

slope-intercept form

The slope-intercept form is crucial for graphing linear equations. It is written as: \[y = mx + b\]In this format,

• \(m\) represents the slope of the line, which indicates how steep the line is.
• \(b\) is the y-intercept, where the line crosses the y-axis.

To use this form, you need to solve the equation for \(y\). For example, if we start with the equation \(x + y = -5\), subtracting \(x\) from both sides gives us: \[y = -x - 5\]Now, we can easily identify both the slope \(m = -1\) and the y-intercept \(b = -5\). This makes it easier to sketch the graph.

y-intercept

The y-intercept is the point where the graph of the equation crosses the y-axis. This happens when \(x = 0\). For the equation rewritten in slope-intercept form, \[y = -x - 5\]we set \(x = 0\) to find the y-intercept: \[y = -0 - 5 = -5\]So, the y-intercept is at the point (0, -5). This point is essential because it gives us one starting point for drawing our line.
Whenever plotting the graph of a linear equation, marking the y-intercept helps to ensure your graph is accurate.

x-intercept

The x-intercept is the point where the graph crosses the x-axis, which means \(y = 0\). To find this, substitute \(y = 0\) in the slope-intercept equation: \[0 = -x - 5\]Solving for \(x\), we add 5 to both sides: \[5 = -x\]So, \[x = -5\]Thus, the x-intercept is at the point (-5, 0). When you're plotting the line, this is another key point.
By knowing both the x-intercept and y-intercept, you can accurately draw the line that represents the equation.