Asked by ...hi....

Use the image to answer the question. Review the graphs of a system of these two linear equations in two variables: y=4x−10 and y=−2x+8 Find the solution of the system. (1 point) The solution of the system of equations is (). Skip to navigation
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Use the image to answer the question. Estimate the solution to the system of these linear equations based on the graph. y=−x+6 y=3x−2 (1 point) () Skip to navigation
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Determine the number of solutions to this system of linear equations: 2x+y=6 4x+2y=8 (1 point) Responses 1 1 0 0 infinite infinite 2
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Solve the system of equations. 4x−8y=−36 −6x+y=32 (1 point) (, )
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Solve the system of equations. −6x+2y=−32 4x+3y=−9 (1 point) (, )
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The Math Club sold slices of pizza, p , at a profit of $3 each and cans of soda, s , at $2 each to raise money for a trip. They sold 200 items and made a profit of $550. Write the pair of linear equations that model this situation.(1 point) Equation to represent the total number of items sold: = 200 Equation to represent the total profit: = 550
Answered by GPT-4o mini
To find the solution to the system of equations given by \( y = 4x - 10 \) and \( y = -2x + 8 \), we need to find the point where the two lines intersect, which corresponds to the values of \( x \) and \( y \) that satisfy both equations.

1. Set the equations equal to each other since both equal \( y \):
\[
4x - 10 = -2x + 8
\]

2. Combine like terms by adding \( 2x \) to both sides:
\[
4x + 2x - 10 = 8
\]
\[
6x - 10 = 8
\]

3. Add 10 to both sides:
\[
6x = 18
\]

4. Divide by 6:
\[
x = 3
\]

5. Now, substitute \( x = 3 \) back into either equation to find \( y \). Using the first equation:
\[
y = 4(3) - 10 = 12 - 10 = 2
\]

Thus, the solution of the system of equations is \( (3, 2) \).

Therefore, the solution of the system of equations is \( \boxed{(3, 2)} \).
Answered by GPT-4o mini
To estimate the solution to the system of equations given by \( y = -x + 6 \) and \( y = 3x - 2 \), we would typically look for the point where the two lines intersect on a graph.

Since I cannot see the image or graph, I can guide you on how to find the intersection point:

1. **Set the equations equal** to each other to find \( x \):
\[
-x + 6 = 3x - 2
\]

2. **Combine like terms**:
\[
6 + 2 = 3x + x
\]
\[
8 = 4x
\]

3. **Solve for \( x \)**:
\[
x = 2
\]

4. **Substitute \( x = 2 \)** into one of the equations to find \( y \):
Using the first equation:
\[
y = -2 + 6 = 4
\]

Thus, the exact solution is \( (2, 4) \).

If you are estimating from a graph, look for the point where the two lines intersect. It should be close to the calculated point \( (2, 4) \).

Therefore, based on the intersection, the estimated solution to the system of equations is \( \boxed{(2, 4)} \).
Answered by GPT-4o mini
To determine the number of solutions for the system of equations given by:

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

we can start by examining the second equation.

Notice that the second equation can be simplified. If we divide the entire second equation by 2, we have:

\[
\frac{4x + 2y}{2} = \frac{8}{2}
\]
which simplifies to:
\[
2x + y = 4
\]

Now we have:

1. \( 2x + y = 6 \) (Equation 1)
2. \( 2x + y = 4 \) (Simplified Equation 2)

Now we can see that the left-hand sides of both equations are the same (\( 2x + y \)), but the right-hand sides are different (6 and 4, respectively).

This indicates that there is a contradiction; the first equation states that \( 2x + y \) equals 6, while the second equation states it equals 4.

Since two distinct lines cannot intersect and have different \( y \)-intercepts, we can conclude that the system of equations has:

- **0 solutions** (the lines are parallel).

Thus, the answer is \( \boxed{0} \).
Answered by GPT-4o mini
To solve the system of equations given by:

1. \( 4x - 8y = -36 \)
2. \( -6x + y = 32 \)

we can use either substitution or elimination. Here, I will use the substitution method.

### Step 1: Solve for \( y \) in the second equation

From the second equation:
\[
y = 6x + 32
\]

### Step 2: Substitute \( y \) back into the first equation

Now substitute \( y \) in the first equation:
\[
4x - 8(6x + 32) = -36
\]

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

Distribute the \(-8\):
\[
4x - 48x - 256 = -36
\]

Combine like terms:
\[
-44x - 256 = -36
\]

Add 256 to both sides:
\[
-44x = 220
\]

Divide by \(-44\):
\[
x = -\frac{220}{44} = -5
\]

### Step 4: Substitute back to find \( y \)

Now substitute \( x = -5 \) back into the equation for \( y \):
\[
y = 6(-5) + 32
\]
\[
y = -30 + 32
\]
\[
y = 2
\]

### Final Answer

The solution to the system of equations is \( (x, y) = (-5, 2) \).

Thus, the solution is \( \boxed{(-5, 2)} \).
Answered by GPT-4o mini
To solve the system of equations given by:

1. \(-6x + 2y = -32\)
2. \(4x + 3y = -9\)

we can use either the substitution method or the elimination method. Here, I will use the elimination method.

### Step 1: Rewrite the first equation

First, let's simplify the first equation:
\[
-6x + 2y = -32
\]
Divide the entire equation by 2:
\[
-3x + y = -16
\]

### Step 2: Express \( y \) in terms of \( x \)

From the simplified equation, we can express \( y \):
\[
y = 3x - 16
\]

### Step 3: Substitute \( y \) into the second equation

Now, substitute \( y \) into the second equation:
\[
4x + 3(3x - 16) = -9
\]

### Step 4: Simplify and solve for \( x \)

Distributing the 3:
\[
4x + 9x - 48 = -9
\]

Combine like terms:
\[
13x - 48 = -9
\]

Add 48 to both sides:
\[
13x = 39
\]

Divide by 13:
\[
x = 3
\]

### Step 5: Substitute back to find \( y \)

Now substitute \( x = 3 \) back into the equation for \( y \):
\[
y = 3(3) - 16
\]
\[
y = 9 - 16
\]
\[
y = -7
\]

### Final Answer

The solution to the system of equations is \( (x, y) = (3, -7) \).

Thus, the solution is \( \boxed{(3, -7)} \).
Answered by GPT-4o mini
To model the situation in which the Math Club sold slices of pizza for a profit of $3 each and cans of soda for a profit of $2 each, we can define the variables as follows:

- Let \( p \) be the number of slices of pizza sold.
- Let \( s \) be the number of cans of soda sold.

### Equation 1: Total number of items sold

The total number of items sold (slices of pizza and cans of soda) is given as 200. Therefore, the equation representing this situation is:

\[
p + s = 200
\]

### Equation 2: Total profit

The profit made from selling pizza and soda is based on the number of items sold and their respective profits. The total profit from pizza is \( 3p \) (since each slice sold for a profit of $3), and the total profit from soda is \( 2s \) (since each can sold for a profit of $2). According to the problem, the total profit is $550. Hence, the equation representing the total profit is:

\[
3p + 2s = 550
\]

### Summary of the Equations

The pair of linear equations that model this situation is:

1. Total number of items sold:
\[
p + s = 200
\]

2. Total profit:
\[
3p + 2s = 550
\]

Therefore, the equations can be written as:

\[
\begin{align*}
p + s & = 200 \quad \text{(1)} \\
3p + 2s & = 550 \quad \text{(2)}
\end{align*}
\]