Solve the following system of equations with the substitution method: 8x-4y=16 y=2x-4

Solve the following system of equations with the substitution method: 8x-4y=16 y=2x-4 Enter your answers as integers or as reduced fractions in the form A/B. If there is no solution, type “DNE” in each blank. If there are an infinite number of solutions, specify their form in the blanks in terms of x.
Solve the following system of equations with the substitution method:

8x-4y=16

y=2x-4

Enter your answers as integers or as reduced fractions in the form A/B. If there is no solution, type “DNE” in each blank. If there are an infinite number of solutions, specify their form in the blanks in terms of x.

The correct answer and explanation is :

The solution to the system of equations is ( x = \frac{y}{2} + 2 ).

Explanation:

We are solving the system of equations using the substitution method. The system is:

[
8x – 4y = 16 \tag{1}
]
[
y = 2x – 4 \tag{2}
]

1. Substitute the expression for (y) from equation (2) into equation (1): Since equation (2) gives ( y = 2x – 4 ), we substitute this expression for (y) into equation (1): [
   8x – 4(2x – 4) = 16
   ]
2. Simplify the equation: Distribute the (-4) over the terms inside the parentheses: [
   8x – 8x + 16 = 16
   ]
3. Simplify further: The (8x) terms cancel out, leaving: [
   16 = 16
   ] This is a true statement, which means the system has an infinite number of solutions.
4. Express the solution in terms of (x): Since the equation (16 = 16) is always true, we know that (x) can take any value, and (y) will be determined based on (x). From equation (2), we have: [
   y = 2x – 4
   ]

Thus, the solution to the system of equations is:

[
x = \frac{y}{2} + 2
]

This means for any value of (y), you can find a corresponding (x) using this relationship, leading to infinitely many solutions.

Image of the system of equations:

Here is the graph that visually represents the system of equations ( 8x – 4y = 16 ) and ( y = 2x – 4 ). The lines show how the solutions of the system are plotted on the coordinate grid. As we see, the lines intersect, indicating that the system has infinitely many solutions, as described earlier.