Solve the system of equations by graphing. x+y=7 2x-4y=8
The Correct Answer and Explanation is:
To solve the system of equations by graphing:
- Equation 1: x+y=7x + y = 7x+y=7
- Equation 2: 2x−4y=82x – 4y = 82x−4y=8
Step 1: Rewrite both equations in slope-intercept form (y = mx + b)
Equation 1: x+y=7y=−x+7x + y = 7 \\ y = -x + 7x+y=7y=−x+7
Equation 2: 2x−4y=8−4y=−2x+8y=12x−22x – 4y = 8 \\ -4y = -2x + 8 \\ y = \frac{1}{2}x – 22x−4y=8−4y=−2x+8y=21x−2
Now we have:
- y=−x+7y = -x + 7y=−x+7
- y=12x−2y = \frac{1}{2}x – 2y=21x−2
Step 2: Graph both lines
To graph y=−x+7y = -x + 7y=−x+7, start at the y-intercept (0, 7). The slope is -1, so for every 1 unit you move to the right, move 1 unit down.
To graph y=12x−2y = \frac{1}{2}x – 2y=21x−2, start at the y-intercept (0, -2). The slope is 1/2, so for every 2 units you move to the right, move 1 unit up.
Step 3: Identify the point of intersection
By graphing both lines, we find that they intersect at the point (6, 1). This means the solution to the system of equations is: (6,1)\boxed{(6, 1)}(6,1)
Explanation
Solving systems of equations by graphing involves plotting each equation on the same coordinate plane and identifying the point where the two lines intersect. This point represents the values of x and y that satisfy both equations at the same time.
The first step is to rearrange each equation into the slope-intercept form y=mx+by = mx + by=mx+b, where m is the slope and b is the y-intercept. This makes it easier to plot the lines since we know where they cross the y-axis and how steep they are.
In this case, the equation x+y=7x + y = 7x+y=7 becomes y=−x+7y = -x + 7y=−x+7, which has a slope of -1 and a y-intercept of 7. That tells us the line starts at (0, 7) and goes downward as x increases. The second equation 2x−4y=82x – 4y = 82x−4y=8 simplifies to y=12x−2y = \frac{1}{2}x – 2y=21x−2, meaning it has a positive slope of 1/2 and a y-intercept of -2. This line starts at (0, -2) and rises slowly as x increases.
After plotting both lines, the intersection point is clearly (6, 1). This is the solution to the system because it is the only point where both equations are true simultaneously. Plugging x = 6 and y = 1 into both original equations confirms the result.
Graphing provides a visual way to understand solutions to linear systems. While it may not always be the most precise method due to limitations in drawing accuracy, it is useful for understanding the relationships between equations and for checking solutions graphically.
