Question

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

Explanation:

Step1: Rewrite first equation in slope - intercept form

The first equation is \(x + 3y=6\). We solve for \(y\):
Subtract \(x\) from both sides: \(3y=-x + 6\).
Divide by 3: \(y=-\frac{1}{3}x + 2\).
To find two points on this line, when \(x = 0\), \(y=-\frac{1}{3}(0)+2 = 2\), so the point is \((0,2)\). When \(x = 3\), \(y=-\frac{1}{3}(3)+2=- 1 + 2=1\), so the point is \((3,1)\).

Step2: Rewrite second equation in slope - intercept form

The second equation is \(4x-6y = 6\). We solve for \(y\):
Subtract \(4x\) from both sides: \(-6y=-4x + 6\).
Divide by \(-6\): \(y=\frac{4}{6}x-1=\frac{2}{3}x - 1\).
To find two points on this line, when \(x = 0\), \(y=\frac{2}{3}(0)-1=-1\), so the point is \((0, - 1)\). When \(x = 3\), \(y=\frac{2}{3}(3)-1=2 - 1 = 1\), so the point is \((3,1)\).

Step3: Find the intersection point

The two lines intersect at the point where the \(x\) and \(y\) values satisfy both equations. From the points we found, both lines pass through \((3,1)\). So the solution to the system of equations is the point of intersection of the two lines.

Answer:

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