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 (2, 2), 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, we need to find 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 (m1) and the y-intercept (b1) for this line:

`\[ \text{Slope (m1)} = (0 - 4)/(6 - 0) = -(2)/(3) \]`

Now, we use the slope-intercept form of a line, y = mx + b, where m is the slope and b is the y-intercept. Substituting one of the given points, say (0,4):

`\[ 4 = -(2)/(3)(0) + b1 \]\[ b1 = 4 \]`

So, the equation of the first line is
`\(y = -(2)/(3)x + 4\)`.

Now, for the second line passing through (0, -2) and (-3, -3), let's find its slope (m2) and y-intercept (b2):

`\[ \text{Slope (m2)} = (-3 - (-2))/((-3) - 0) = (-1)/(-3) = (1)/(3) \]`

Using the point-slope form,
`\(y - y1 = m(x - x1)\)`, and substituting the point (0, -2):

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

Now, equate the two equations to find the intersection:

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

Combine like terms:

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

Now, substitute x = 2 into either equation (let's use the first one):

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

So, the solution to the system is the ordered pair (2, 2).