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Chapter 3: Problem 47

In the following exercises, graph using the intercepts. $$ x+y=-3 $$

Short Answer

Expert verified
The x-intercept is (-3, 0) and the y-intercept is (0, -3). Plot these points and draw a line through them.

Step by step solution

01

Identify the x-intercept

To find the x-intercept, set y to 0 in the equation and solve for x:\[ x + y = -3 \]Set \( y = 0 \):\[ x + 0 = -3 \]Thus, \( x = -3 \). The x-intercept is \((-3, 0)\).
02

Identify the y-intercept

To find the y-intercept, set x to 0 in the equation and solve for y:\[ x + y = -3 \]Set \( x = 0 \):\[ 0 + y = -3 \]Thus, \( y = -3 \). The y-intercept is \((0, -3)\).
03

Plot the intercepts

On a coordinate plane, plot the points \((-3, 0)\) and \((0, -3)\). These are the x- and y-intercepts, respectively.
04

Draw the line

Draw a straight line passing through both plotted points. This line is the graph of the equation \( x + y = -3 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

x-intercept
The x-intercept is the point where a graph crosses the x-axis. To find this point for a given equation, we set y to 0 and solve for x. For example, with the equation \( x + y = -3 \), we set \( y = 0 \):\[ x + 0 = -3 \]Solving for x, we get \( x = -3 \). Therefore, the x-intercept is \( (-3, 0) \). This point helps us understand where the line will touch or cross the x-axis on the coordinate plane.
y-intercept
The y-intercept is the point where a graph crosses the y-axis. To find this point for a given equation, we set x to 0 and solve for y. For the equation \( x + y = -3 \), we set \( x = 0 \):\[ 0 + y = -3 \]Solving for y, we get \( y = -3 \). Hence, the y-intercept is \( (0, -3) \). This point shows where the line will touch or cross the y-axis, giving us a clear positional reference on the coordinate plane.
coordinate plane
The coordinate plane is a two-dimensional surface formed by two perpendicular lines called axes. The horizontal axis is the x-axis, and the vertical axis is the y-axis. Each point on this plane is identified by a pair of numerical coordinates \((x,y)\), where\(x \) is the distance from the y-axis, and \(y \) is the distance from the x-axis. This plane helps us visualize and locate points and graph linear equations. For graphing, intercepts provide crucial reference points to draw the line of the equation.
plotting points
Plotting points on the coordinate plane involves marking the position of coordinates \((x, y)\). Each point corresponds to a pair of numbers that dictates its location on the plane. For instance, to plot the intercepts from the equation \( x + y = -3 \), the points are \((-3, 0)\) and \((0, -3)\). Mark \(-3\) on the x-axis and plot the point by moving vertically to \(0\). Similarly, mark \(0\) on the y-axis and move horizontally to \(-3\). Plotting these points and connecting them with a straight line graphically represents the linear equation.

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