Problem 25 - Elementary Algebra

Question

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing. $$ \left\\{\begin{array}{l} x+y=-5 \\ x-y=3 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \( (-1, -4) \).

Step 1 - Write the equations in slope-intercept form

The first equation is: \[ x + y = -5 \]To get it in slope-intercept form (\( y = mx + b \)), solve for \( y \): \[ y = -x - 5 \]The second equation is: \[ x - y = 3 \]Solve for \( y \): \[ y = x - 3 \]

Step 2 - Graph the first equation

Graph the line \( y = -x - 5 \) on the coordinate plane. This equation has a slope of -1 and a y-intercept of -5. Start at the point (0, -5) and use the slope to find another point. From (0, -5), go down 1 unit and right 1 unit to plot another point at (1, -6).

Step 3 - Graph the second equation

Graph the line \( y = x - 3 \). This equation has a slope of 1 and a y-intercept of -3. Start at the point (0, -3) and use the slope to find another point. From (0, -3), go up 1 unit and right 1 unit to plot another point at (1, -2).

Step 4 - Find the point of intersection

The point where the two lines intersect is the solution to the system of equations. From the graphs, observe the intersection point. The lines intersect at the point (-1, -4).

Step 5 - Verify the solution

Verify the solution by substituting \( x = -1 \) and \( y = -4 \) back into the original equations. For the first equation: \[ -1 + (-4) = -5 \] This is true. For the second equation: \[ -1 - (-4) = -1 + 4 = 3 \] This is also true.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that yield a straight line when graphed on a coordinate plane. The general form is \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. A key feature of linear equations is that their graph results in a straight line.

Slope-Intercept Form

The slope-intercept form of a linear equation is particularly useful for graphing. It is expressed as \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) is the y-intercept. The slope indicates how steep the line is, and the y-intercept tells us where the line crosses the y-axis.
For example, if we have an equation like \(y = -x - 5\), the slope \(m\) is -1, and the y-intercept \(b\) is -5. This means that the line will slant downwards and cross the y-axis at (0, -5).

Graphing Systems of Equations

Graphing systems of equations involves plotting multiple equations on the same coordinate plane. Each equation represents a line. The solution to the system is the point where the lines intersect.
For instance, in the given problem, we graph \(x + y = -5\) and \(x - y = 3\) by first rewriting them in slope-intercept form as \(y = -x - 5\) and \(y = x - 3\) respectively.
We then plot each line using their slopes and y-intercepts. The intersection point is where we can derive the values of \(x\) and \(y\) that satisfy both equations simultaneously.

Point of Intersection

The point of intersection is where two or more lines on a graph meet. This point represents the solution to a system of linear equations.
In our example, the lines represented by \(y = -x - 5\) and \(y = x - 3\) intersect at (-1, -4). This means \(x = -1\) and \(y = -4\) is the solution to the system.
Always verify by plugging the coordinates back into the original equations to make sure they satisfy both. For the first equation, \(-1 + (-4) = -5\), which is true, and for the second, \(-1 + 4 = 3\), which is also true. This affirms the intersection point as the correct solution.