Question

solve the system by graphing. \

$$\begin{cases} 6x + 3y = -21 \\\\ -2x + y = 1 \\end{cases}$$

use the graphing tool to graph the system. click to enlarge graph

Explanation:

Step1: Rewrite equations in slope - intercept form

For the equation \(6x + 3y=-21\), we solve for \(y\):
Subtract \(6x\) from both sides: \(3y=-6x - 21\)
Divide by 3: \(y=-2x - 7\)

For the equation \(-2x + y = 1\), we solve for \(y\):
Add \(2x\) to both sides: \(y = 2x+1\)

Step2: Find intercepts for \(y=-2x - 7\)

• \(y\) - intercept: When \(x = 0\), \(y=-7\). So the point is \((0,-7)\)
• \(x\) - intercept: When \(y = 0\), \(0=-2x - 7\), \(2x=-7\), \(x=-\frac{7}{2}=-3.5\). So the point is \((-3.5,0)\)

Step3: Find intercepts for \(y = 2x + 1\)

• \(y\) - intercept: When \(x = 0\), \(y = 1\). So the point is \((0,1)\)
• \(x\) - intercept: When \(y = 0\), \(0=2x + 1\), \(2x=-1\), \(x=-\frac{1}{2}=-0.5\). So the point is \((-0.5,0)\)

Step4: Graph the lines

Plot the points for each line and draw the lines. The intersection point of the two lines \(y=-2x - 7\) and \(y = 2x + 1\) is the solution.
To find the intersection algebraically (to confirm), set \(-2x - 7=2x + 1\)
Add \(2x\) to both sides: \(-7 = 4x+1\)
Subtract 1: \(-8 = 4x\)
Divide by 4: \(x=-2\)
Substitute \(x = - 2\) into \(y = 2x + 1\): \(y=2(-2)+1=-4 + 1=-3\)

Answer:

The solution to the system is \(x=-2,y = - 3\) or the ordered pair \((-2,-3)\)