Question

12. Solve the following system of
equations by graphing.
y = −x + 5
y = 2x − 4

Answer 1

13. How many solutions does the
following system of equations have?

−x + 2y = 5
2x − 4y = −8

Answer 2

To solve this system of equations by graphing, we first need to graph each equation on the same coordinate plane and find the point where the two lines intersect. This point will be the solution to the system of equations.

First, let's graph the equation y = -x + 5:
This equation is in slope-intercept form y = mx + b, where m is the slope and b is the y-intercept. The slope of this equation is -1 (coefficient of x) and the y-intercept is 5.

Plot the y-intercept (0,5) on the y-axis and use the slope to find another point by going down 1 unit and to the right 1 unit from the y-intercept. This gives us the points (1,4) and (-1,6). Connect these points to draw a line.

Next, let's graph the equation y = 2x - 4:
This equation is also in slope-intercept form with a slope of 2 and a y-intercept of -4.

Plot the y-intercept (0,-4) on the y-axis and use the slope to find another point by going up 2 units and to the right 1 unit from the y-intercept. This gives us the points (1,-2) and (-1,-6). Connect these points to draw a line.

Now, you should have two lines on the coordinate plane. The point where they intersect is the solution to the system of equations. In this case, the intersection point is (3,1).

Therefore, the solution to the system of equations y = -x + 5 and y = 2x - 4 is x = 3, y = 1.

Answer 3

To determine how many solutions the system of equations has, we need to graph the two equations and see if they intersect at a single point, have infinitely many solutions (the same line), or have no solutions (parallel lines).

First, let's rewrite the given system of equations in slope-intercept form:
1. -x + 2y = 5
Rearrange the equation to solve for y:
2y = x + 5
y = 1/2x + 5/2

2. 2x - 4y = -8
Rearrange the equation to solve for y:
-4y = -2x - 8
y = 1/2x + 2

Now, let's graph the two equations on the same coordinate plane. The equations have the same slope (1/2) but different y-intercepts (5/2 and 2), which means the lines are parallel.

Since parallel lines never intersect, the system of equations has no solution. The equations represent two lines that are parallel and do not cross each other.