Solve the system of linear equations by graphing: Y – 4x = 3x + y = 7. Use the graphing tool to graph the system. Click to enlarge graph. Select the correct choice below and, if necessary, fill in the answer box to complete your choice: The solution of the system is (Type an ordered pair). There are infinitely many solutions: ((x,y) = (4x) or ((x,y) = (x+7). There is no solution: (0) or (0).
The Correct Answer and Explanation is:
To solve the system of linear equations by graphing, begin by rewriting the system clearly and then putting each equation into slope-intercept form (y = mx + b):
Given system of equations:
- y−4x=3xy – 4x = 3xy−4x=3x
- y=7y = 7y=7
Step 1: Simplify the first equation
Start with:y−4x=3xy – 4x = 3xy−4x=3x
Add 4x4x4x to both sides:y=7xy = 7xy=7x
So the system becomes:
- y=7xy = 7xy=7x
- y=7y = 7y=7
Step 2: Graph the equations
- Equation 1: y=7xy = 7xy=7x
- This is a straight line with a slope of 7 and y-intercept of 0.
- Passes through (0, 0), (1, 7), (–1, –7).
- Equation 2: y=7y = 7y=7
- This is a horizontal line crossing the y-axis at 7.
- Passes through points like (0, 7), (1, 7), (2, 7).
Step 3: Find the intersection
To find the solution graphically, look for the point of intersection of the two lines.
Set the two equations equal to each other:7x=77x = 77x=7
Solve for xxx:x=1x = 1x=1
Substitute back into either equation:y=7(1)=7y = 7(1) = 7y=7(1)=7
So, the solution is:(1,7)\boxed{(1, 7)}(1,7)
Answer Choice:
✅ The solution of the system is (1,7)\boxed{(1, 7)}(1,7)
Explanation (Like in Textbooks):
A system of linear equations can be solved graphically by finding the point where the graphs of the equations intersect. Each equation represents a straight line. The point at which the two lines cross is the solution to the system, because it satisfies both equations.
In this case, the first equation simplifies to y=7xy = 7xy=7x, and the second equation is a constant line, y=7y = 7y=7. By graphing both lines on the same coordinate plane, we observe that they intersect at the point (1,7)(1, 7)(1,7). This means that (1,7)(1, 7)(1,7) satisfies both equations, and thus, is the unique solution to the system. Since the lines intersect at a single point, the system is consistent and independent.
