Question

The graphed line represents the first equation in a system of equations. If the second line passes through the points (0, -2) and (-3, -3), find the solution to the system. Write your answer as an ordered pair, (x, y).

Answer 1

The solution to the system of equations is (18, -8), representing the point of intersection between the lines given by the points (0,4) and (6,0) and passing through (0, -2) and (-3, -3).

To find the solution to the system of equations, we need to determine the point of intersection between the two lines. The first line is represented by the points (0,4) and (6,0). Let's find the slope
`(\(m_1\))` and the y-intercept
`(\(b_1\))` for this line:

`\[ m_1 = (0 - 4)/(6 - 0) = -(2)/(3) \]`

Using the point-slope form
`(\(y - y_1 = m(x - x_1)\))`, substitute one of the given points, say (0,4):

`\[ y - 4 = -(2)/(3)(x - 0) \]`

Simplify this to the slope-intercept form y = mx + b:

`\[ y = -(2)/(3)x + 4 \]`

Now, for the second line passing through (0, -2) and (-3, -3), let's find its slope
`(\(m_2\))` and y-intercept
`(\(b_2\))`:

`\[ m_2 = (-3 - (-2))/((-3) - 0) = (-1)/(3) \]`

Using the point-slope form:

`\[ y + 2 = (-1)/(3)(x - 0) \]`

Simplify this to the slope-intercept form:

`\[ y = (-1)/(3)x - 2 \]`

Now, set the two equations equal to each other:

`\[ -(2)/(3)x + 4 = (-1)/(3)x - 2 \]`

Combine like terms:

`\[ -(2)/(3)x + (1)/(3)x = -2 - 4 \]\[ -(1)/(3)x = -6 \]\[ x = 18 \]`

Now, substitute x = 18 into either equation (let's use the first one):

`\[ y = -(2)/(3)(18) + 4 \]\[ y = -12 + 4 \]\[ y = -8 \]`

So, the solution to the system is the ordered pair
`\((18, -8)\)`.