Question

2x + 3y = 6

-4x + 8y = 8

-5x - 10y = -20

6x - 2y = -12

Explanation:

Since the problem seems to be about graphing linear equations (finding x - intercept and y - intercept to plot the line), we can solve for each equation's intercepts. Let's take one equation at a time, say \(2x + 3y=6\)

Step 1: Find y - intercept (x = 0)

Substitute \(x = 0\) into the equation \(2x+3y = 6\)
\(2(0)+3y=6\)
\(3y = 6\)
Divide both sides by 3: \(y=\frac{6}{3}=2\). So the y - intercept is \((0,2)\)

Step 2: Find x - intercept (y = 0)

Substitute \(y = 0\) into the equation \(2x + 3y=6\)
\(2x+3(0)=6\)
\(2x=6\)
Divide both sides by 2: \(x=\frac{6}{2}=3\). So the x - intercept is \((3,0)\)

We can plot the points \((0,2)\) and \((3,0)\) and draw a line through them.

Let's do the same for \( - 4x+8y = 8\)

Step 1: Find y - intercept (x = 0)

Substitute \(x = 0\) into \(-4x + 8y=8\)
\(-4(0)+8y=8\)
\(8y=8\)
\(y = 1\). So y - intercept is \((0,1)\)

Step 2: Find x - intercept (y = 0)

Substitute \(y = 0\) into \(-4x+8y = 8\)
\(-4x+8(0)=8\)
\(-4x=8\)
\(x=\frac{8}{-4}=-2\). So x - intercept is \((-2,0)\)

For \(-5x-10y=-20\)

Step 1: Find y - intercept (x = 0)

Substitute \(x = 0\) into \(-5x-10y=-20\)
\(-5(0)-10y=-20\)
\(-10y=-20\)
\(y=\frac{-20}{-10} = 2\). So y - intercept is \((0,2)\)

Step 2: Find x - intercept (y = 0)

Substitute \(y = 0\) into \(-5x-10y=-20\)
\(-5x-10(0)=-20\)
\(-5x=-20\)
\(x=\frac{-20}{-5}=4\). So x - intercept is \((4,0)\)

For \(6x-2y=-12\)

Step 1: Find y - intercept (x = 0)

Substitute \(x = 0\) into \(6x-2y=-12\)
\(6(0)-2y=-12\)
\(-2y=-12\)
\(y = 6\). So y - intercept is \((0,6)\)

Step 2: Find x - intercept (y = 0)

Substitute \(y = 0\) into \(6x-2y=-12\)
\(6x-2(0)=-12\)
\(6x=-12\)
\(x=-2\). So x - intercept is \((-2,0)\)

If we consider the equation \(-5x - 10y=-20\), we can also simplify it first. Divide the entire equation by \(-5\): \(x + 2y=4\)

Step 1: Find y - intercept (x = 0)

\(0+2y=4\)
\(y = 2\) (same as before)

Step 2: Find x - intercept (y = 0)

\(x+2(0)=4\)
\(x = 4\) (same as before)

To graph each line, plot the x - intercept and y - intercept points and draw a straight line through them.

If the question was to graph all these lines, the process for each line is as above.

Answer:

For \(2x + 3y=6\): Intercepts \((0,2)\) and \((3,0)\)
For \(-4x + 8y=8\): Intercepts \((0,1)\) and \((-2,0)\)
For \(-5x-10y=-20\): Intercepts \((0,2)\) and \((4,0)\)
For \(6x-2y=-12\): Intercepts \((0,6)\) and \((-2,0)\)

(To graph, plot these points and draw lines through them for each equation)