Question

5 solve the system of linear equations by graphing. (3 pts) ( x + 3y = 6 ) ( 4x - 6y = 6 )

Explanation:

Step1: Rewrite equations in slope - intercept form ($y = mx + b$)

For the first equation \(x + 3y=6\):
Subtract \(x\) from both sides: \(3y=-x + 6\)
Divide by 3: \(y=-\frac{1}{3}x + 2\)

For the second equation \(4x-6y = 6\):
Subtract \(4x\) from both sides: \(-6y=-4x + 6\)
Divide by \(-6\): \(y=\frac{2}{3}x-1\)

Step2: Find two points for each line

For \(y =-\frac{1}{3}x + 2\)

• When \(x = 0\), \(y=-\frac{1}{3}(0)+2 = 2\). So one point is \((0,2)\).
• When \(x = 3\), \(y=-\frac{1}{3}(3)+2=-1 + 2=1\). So another point is \((3,1)\).

For \(y=\frac{2}{3}x-1\)

• When \(x = 0\), \(y=\frac{2}{3}(0)-1=-1\). So one point is \((0,-1)\).
• When \(x = 3\), \(y=\frac{2}{3}(3)-1=2 - 1=1\). So another point is \((3,1)\).

Step3: Graph the lines and find intersection

The two lines intersect at the point \((3,1)\) since both lines pass through \((3,1)\).

Answer:

The solution to the system of linear equations is \(x = 3,y = 1\) or the ordered pair \((3,1)\).