Question

2.
$x + 3y = 6$
$2x - 3y = 3$
solution:
coordinate grid for plotting

Explanation:

Step1: Solve the system of equations

We have the system:

$$
LATEXBLOCK0
$$

Add the two equations to eliminate \(y\):
\((x + 3y)+(2x - 3y)=6 + 3\)
\(x+2x+3y - 3y=9\)
\(3x=9\)
Divide both sides by 3: \(x = 3\)

Step2: Substitute \(x = 3\) into the first equation

Substitute \(x = 3\) into \(x + 3y = 6\):
\(3+3y = 6\)
Subtract 3 from both sides: \(3y=6 - 3=3\)
Divide both sides by 3: \(y = 1\)

Step3: Find the intersection point

The solution to the system is the point \((3,1)\), which is the intersection of the two lines represented by the equations. To graph, we can also find the intercepts for each line:

For \(x + 3y = 6\):

• When \(x = 0\), \(3y=6\Rightarrow y = 2\), so the y - intercept is \((0,2)\)
• When \(y = 0\), \(x=6\), so the x - intercept is \((6,0)\)

For \(2x - 3y = 3\):

• When \(x = 0\), \(-3y=3\Rightarrow y=- 1\), so the y - intercept is \((0,-1)\)
• When \(y = 0\), \(2x=3\Rightarrow x=\frac{3}{2}=1.5\), so the x - intercept is \((1.5,0)\)

Plot these points and draw the lines. The lines will intersect at \((3,1)\)

Answer:

The solution to the system of equations \(

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

\) is \(x = 3\) and \(y = 1\) (or the ordered pair \((3,1)\)). When graphing, the two lines intersect at the point \((3,1)\).