what is the graph of the system $y = -2x + 3$ and $2x + 4y = 8$?
what is the graph of the system $y = -2x + 3$ and $2x + 4y = 8$?
Answer
Explanation:
Step1: Rewrite the second equation in slope - intercept form
The second equation is (2x + 4y=8). We want to solve for (y) to get it in the form (y = mx + b) (where (m) is the slope and (b) is the y - intercept). First, subtract (2x) from both sides: (4y=-2x + 8). Then, divide each term by 4: (y=\frac{-2x + 8}{4}=-\frac{1}{2}x + 2). The first equation is (y=-2x + 3).
Step2: Analyze the slopes and y - intercepts
- For the first line (y=-2x + 3): The slope (m_1=-2) and the y - intercept (b_1 = 3) (so it crosses the y - axis at ((0,3))).
- For the second line (y =-\frac{1}{2}x+2): The slope (m_2=-\frac{1}{2}) and the y - intercept (b_2 = 2) (so it crosses the y - axis at ((0,2))).
Now, let's analyze the options:
- The first line has a steeper slope (since (\vert - 2\vert>\vert-\frac{1}{2}\vert)) and a higher y - intercept.
- The second line has a less steep slope and a lower y - intercept.
Looking at the graphs, the third option (the one with the two lines where one has a y - intercept at 3 and a steeper slope, and the other has a y - intercept at 2 and a less steep slope) matches the equations.