Question

find the x and y intercepts, then graph the line.
$2x + 3y = 6$
$-4x + 8y = 8$
$-5x - 10y = -20$
$6x - 2y = -12$

Explanation:

For the equation \(2x + 3y = 6\)

Step1: Find x-intercept (set \(y = 0\))

Substitute \(y = 0\) into \(2x + 3y = 6\):
\(2x + 3(0) = 6\)
\(2x = 6\)
\(x = \frac{6}{2} = 3\)
So, x-intercept is \((3, 0)\).

Step2: Find y-intercept (set \(x = 0\))

Substitute \(x = 0\) into \(2x + 3y = 6\):
\(2(0) + 3y = 6\)
\(3y = 6\)
\(y = \frac{6}{3} = 2\)
So, y-intercept is \((0, 2)\).

For the equation \(-4x + 8y = 8\)

Step1: Find x-intercept (set \(y = 0\))

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

Step2: Find y-intercept (set \(x = 0\))

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

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

Step1: Find x-intercept (set \(y = 0\))

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

Step2: Find y-intercept (set \(x = 0\))

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

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

Answer:

Step1: Find x-intercept (set \(y = 0\))

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

Step2: Find y-intercept (set \(x = 0\))

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

Graphing (Brief):

• For \(2x + 3y = 6\): Plot \((3, 0)\) and \((0, 2)\), then draw a line through them.
• For \(-4x + 8y = 8\): Plot \((-2, 0)\) and \((0, 1)\), then draw a line through them.
• For \(-5x - 10y = -20\): Plot \((4, 0)\) and \((0, 2)\), then draw a line through them.
• For \(6x - 2y = -12\): Plot \((-2, 0)\) and \((0, 6)\), then draw a line through them.

Final Answers (Intercepts):

• \(2x + 3y = 6\): x-intercept \(\boldsymbol{(3, 0)}\), y-intercept \(\boldsymbol{(0, 2)}\)
• \(-4x + 8y = 8\): x-intercept \(\boldsymbol{(-2, 0)}\), y-intercept \(\boldsymbol{(0, 1)}\)
• \(-5x - 10y = -20\): x-intercept \(\boldsymbol{(4, 0)}\), y-intercept \(\boldsymbol{(0, 2)}\)
• \(6x - 2y = -12\): x-intercept \(\boldsymbol{(-2, 0)}\), y-intercept \(\boldsymbol{(0, 6)}\)