Question

How many solutions does the system of equations have?
8x - 6y = 48
8x + 6y = 48

a. none
b. two
c. one
d. infinite

Answer 1

Solve the system of two equations using elimination with multipliation.
4x - 6y = -48
-2x + 4y = 30

A. (-3, 6)
B. (3, 9)
C. (-21, -6)
D. (-33, -9)

Answer 2

solve the system of equations.
3x - 4y = 40
5x - 5y = 55

a. (4, -7)
b. (18, 7)
c. (14, 3)
d. (8, -3)

Answer 3

how many solutions does the system have?
12x - 15y = 18
4x - 5y = 6

a. two solutions
b. no solution
c. one solution
d. infinitely many solutions

Answer 4

To determine how many solutions the system of equations has, we can analyze the given equations:

1. \( 8x - 6y = 48 \)
2. \( 8x + 6y = 48 \)

Let's rewrite these equations and look for solutions.

From the first equation, we can isolate \(y\):
\[
8x - 6y = 48 \implies -6y = 48 - 8x \implies y = \frac{8x - 48}{6} = \frac{4x - 24}{3}
\]

From the second equation, we can isolate \(y\) as well:
\[
8x + 6y = 48 \implies 6y = 48 - 8x \implies y = \frac{48 - 8x}{6} = 8 - \frac{4x}{3}
\]

Now we have two equations for \(y\):
1. \( y = \frac{4x - 24}{3} \)
2. \( y = 8 - \frac{4x}{3} \)

To find the intersection, set the two equations for \(y\) equal to each other:
\[
\frac{4x - 24}{3} = 8 - \frac{4x}{3}
\]

Multiply by 3 to eliminate the fractions:
\[
4x - 24 = 24 - 4x
\]

Now combine like terms:
\[
4x + 4x = 24 + 24 \implies 8x = 48 \implies x = 6
\]

Substituting \(x = 6\) back into one of the original equations to find \(y\). Let's use the first one:
\[
8(6) - 6y = 48 \implies 48 - 6y = 48 \implies -6y = 0 \implies y = 0
\]

Thus, we have a single solution: \( (x, y) = (6, 0) \).

Therefore, the answer is:
c. one

Answer 5

To solve the system of equations using elimination, we have the following two equations:

1. \( 4x - 6y = -48 \)
2. \( -2x + 4y = 30 \)

We'll multiply the second equation by 2 so that the coefficients of \(x\) will match in both equations allowing us to eliminate \(x\):

\[
2(-2x + 4y) = 2(30) \implies -4x + 8y = 60
\]

Now we have the modified system:

1. \( 4x - 6y = -48 \)
2. \( -4x + 8y = 60 \)

Next, we can add both equations together:

\[
(4x - 6y) + (-4x + 8y) = -48 + 60
\]
This simplifies to:

\[
0x + 2y = 12 \implies 2y = 12 \implies y = 6
\]

Now that we have \(y\), we can substitute \(y = 6\) back into one of the original equations to find \(x\). Let's use the first equation:

\[
4x - 6(6) = -48
\]
\[
4x - 36 = -48
\]
\[
4x = -48 + 36
\]
\[
4x = -12 \implies x = -3
\]

So, the solution to the system of equations is:

\[
(x, y) = (-3, 6)
\]

Thus, the answer is:
**A. (-3, 6)**.

Answer 6

To solve the system of equations:

1. \(3x - 4y = 40\)
2. \(5x - 5y = 55\)

We can use the method of elimination or substitution. Let's use elimination in this case.

First, we can simplify the second equation:
\[
5x - 5y = 55 \implies x - y = 11 \implies x = y + 11
\]

Now we can substitute \(x = y + 11\) into the first equation:

\[
3(y + 11) - 4y = 40
\]

Distribute the \(3\):
\[
3y + 33 - 4y = 40
\]

Combine like terms:
\[
-y + 33 = 40
\]
\[
-y = 40 - 33 \implies -y = 7 \implies y = -7
\]

Now substitute \(y = -7\) back into the equation \(x = y + 11\) to find \(x\):
\[
x = -7 + 11 = 4
\]

Thus, the solution to the system of equations is:
\[
(x, y) = (4, -7)
\]

So, the answer is:
**a. (4, -7)**.

Answer 7

To determine how many solutions the system of equations has, we can analyze the given equations:

1. \( 12x - 15y = 18 \)
2. \( 4x - 5y = 6 \)

To check if these equations are equivalent or if one is a multiple of the other, we can simplify each equation.

Let's simplify the second equation \(4x - 5y = 6\) by multiplying it by 3:

\[
3(4x - 5y) = 3(6)
\]

This gives us:

\[
12x - 15y = 18
\]

Now, we can see that both equations are equivalent:

1. \( 12x - 15y = 18 \)
2. \( 12x - 15y = 18 \) (after multiplying the second equation by 3)

Since both equations represent the same line, there are infinitely many solutions to this system (every point on the line is a solution).

Therefore, the answer is:
**d. infinitely many solutions.**