Question

question 20
two equations are graphed on the same coordinate plane. one line goes through the points (0,2) and (4,4). the other goes through the points (0,6) and (4,0).
do these lines intersect? if so, what is the solution to the system? justify your answer.
enter your response here

Explanation:

Step1: Find slope of first line

The slope $m_1$ of a line passing through $(x_1,y_1)=(0,2)$ and $(x_2,y_2)=(4,4)$ is given by $m_1=\frac{y_2 - y_1}{x_2 - x_1}=\frac{4 - 2}{4 - 0}=\frac{2}{4}=\frac{1}{2}$. Using the slope - intercept form $y=mx + b$ and the point $(0,2)$ (where $b = 2$), the equation of the first line is $y=\frac{1}{2}x+2$.

Step2: Find slope of second line

The slope $m_2$ of a line passing through $(x_1,y_1)=(0,6)$ and $(x_2,y_2)=(4,0)$ is $m_2=\frac{y_2 - y_1}{x_2 - x_1}=\frac{0 - 6}{4 - 0}=\frac{-6}{4}=-\frac{3}{2}$. Using the slope - intercept form and the point $(0,6)$ (where $b = 6$), the equation of the second line is $y=-\frac{3}{2}x + 6$.

Step3: Set the two equations equal

Set $\frac{1}{2}x+2=-\frac{3}{2}x + 6$. Add $\frac{3}{2}x$ to both sides: $\frac{1}{2}x+\frac{3}{2}x+2=-\frac{3}{2}x+\frac{3}{2}x + 6$, which simplifies to $2x+2 = 6$. Subtract 2 from both sides: $2x=4$. Divide by 2: $x = 2$.

Step4: Find the y - value

Substitute $x = 2$ into the first equation $y=\frac{1}{2}(2)+2=1 + 2=3$.

Answer:

Yes, the lines intersect. The solution to the system is $(2,3)$.