Question

Use the substitution method to solve the system 2x+3y=4
, −4x−6y=−8
.(1 point)
Responses

(2−1)
left parenthesis 2 comma negative 1 right parenthesis

infinite solutions
infinite solutions

(−1,2)
left parenthesis negative 1 comma 2 right parenthesis

no solution

Answer 1

Use the substitution method to solve the system x=y−4
, x+8y=2
.(1 point)
Responses

(23,−313)
left parenthesis Start Fraction 2 over 3 End Fraction comma negative 3 Start Fraction 1 over 3 End Fraction right parenthesis

infinite solutions
infinite solutions

no solution
no solution

(−313,23)

Answer 2

Use the substitution method to solve the system 2.5x+y=−2
, 3x+2y=0
.(1 point)
Responses

(3,−2)
left parenthesis 3 comma negative 2 right parenthesis

(−2,3)
left parenthesis negative 2 comma 3 right parenthesis

infinite solutions
infinite solutions

no solution

Answer 3

To solve the system of equations using the substitution method, let's start with the equations provided:

1. \(2x + 3y = 4\) (Equation 1)
2. \(-4x - 6y = -8\) (Equation 2)

### Step 1: Simplify Equation 2
Notice that Equation 2 can be simplified. If we divide the entire equation by -2, we get:
\[
2x + 3y = 4
\]
This is the same as Equation 1.

### Step 2: Analyze the Equations
Since both equations are identical, this means that they represent the same line. Therefore, rather than having a unique solution, this system of equations has infinitely many solutions along the line represented by either equation.

### Conclusion
The correct response is **infinite solutions**.

Answer 4

To solve the system of equations using the substitution method, let's first rewrite the equations:

1. \( x = y - 4 \) (Equation 1)
2. \( x + 8y = 2 \) (Equation 2)

### Step 1: Substitute Equation 1 into Equation 2

Now we substitute \( x \) from Equation 1 into Equation 2.

\[
(y - 4) + 8y = 2
\]

### Step 2: Simplify and Solve for \( y \)

Combine like terms:

\[
y - 4 + 8y = 2
\]
\[
9y - 4 = 2
\]

Now, add 4 to both sides:

\[
9y = 6
\]

Finally, divide by 9:

\[
y = \frac{6}{9} = \frac{2}{3}
\]

### Step 3: Substitute \( y \) back into Equation 1 to find \( x \)

Now, substitute \( y \) back into Equation 1:

\[
x = y - 4
\]
\[
x = \frac{2}{3} - 4
\]
Convert 4 to a fraction with a denominator of 3:

\[
x = \frac{2}{3} - \frac{12}{3} = \frac{2 - 12}{3} = \frac{-10}{3}
\]

### Conclusion

The solution to the system is:
\[
\left( \frac{-10}{3}, \frac{2}{3} \right)
\]

It appears there was a small oversight in the original equations or potential responses as none of the proposed responses exactly match \( \left( \frac{-10}{3}, \frac{2}{3} \right) \). The correct coordinates are based on the derived equations, so please double-check the provided options if they might have included some typographical errors.

If we could have another look at the potential responses, it would certainly help clarify the final answer. In terms of what's provided, none are correct based on the calculations shown.

Answer 5

Let's solve the system of equations using the substitution method:

1. \(2.5x + y = -2\) (Equation 1)
2. \(3x + 2y = 0\) (Equation 2)

### Step 1: Solve Equation 1 for \(y\)

We can solve Equation 1 for \(y\):

\[
y = -2 - 2.5x \quad \text{(Equation 3)}
\]

### Step 2: Substitute Equation 3 into Equation 2

Now we can substitute Equation 3 into Equation 2:

\[
3x + 2(-2 - 2.5x) = 0
\]

### Step 3: Simplify and solve for \(x\)

Distributing \(2\):

\[
3x - 4 - 5x = 0
\]

Combine like terms:

\[
-2x - 4 = 0
\]

Now, add 4 to both sides:

\[
-2x = 4
\]

Divide by -2:

\[
x = -2
\]

### Step 4: Substitute \(x\) back into Equation 3 to find \(y\)

Now substitute \(x = -2\) back into Equation 3:

\[
y = -2 - 2.5(-2)
\]

Calculating \(2.5 \times -2\):

\[
y = -2 + 5 = 3
\]

### Conclusion

The solution to the system is:

\[
\left(-2, 3\right)
\]

Thus, the correct response from the provided options is **\((-2, 3)\)**.