Asked by I Like Fighting Babies

Solve the system of two linear equations. −3x−4y=−23 3x−5y=5 (1 point) Responses (5,−2) left parenthesis 5 comma negative 2 right parenthesis (5, 2) left parenthesis 5 comma 2 right parenthesis (−5, 2) left parenthesis negative 5 comma 2 right parenthesis (−5,−2)
Answered by I Like Fighting Babies
How many solutions does the system of equations have? 8x−6y=48 8x+6y=48 (1 point) Responses one one two two none none infinite
Answered by I Like Fighting Babies
Solve the system of two equations using elimination with multiplication. 4x−6y=−48 −2x+4y=30 (1 point) Responses (3, 9) left parenthesis 3 comma 9 right parenthesis (−21,−6) left parenthesis negative 21 comma negative 6 right parenthesis (−33,−9) left parenthesis negative 33 comma negative 9 right parenthesis (−3, 6)
Answered by I Like Fighting Babies
Solve the system of equations. 3x−4y=40 5x−5y=55 (1 point) Responses (8,−3) left parenthesis 8 comma negative 3 right parenthesis (14, 3) left parenthesis 14 comma 3 right parenthesis (4,−7) left parenthesis 4 comma negative 7 right parenthesis (18, 7)
Answered by I Like Fighting Babies
How many solutions does the system have? 12x−15y=18 4x−5y=6 (1 point) Responses infinitely many solutions infinitely many solutions one solution one solution two solutions two solutions no solution
Answered by I Like Fighting Babies
The senior class at Hartford High School is planning their prom. They’ve narrowed the location down to two choices. The Regal Hotel charges, C , $500 for the ballroom rental fee and $75 per student, s , for the meal. The Greenhill Country Club charges, C , $1,000 for the ballroom rental fee and $65 per student, s , for the meal. Create a pair of linear equations that represent this real-world problem.(1 point) Responses Regal Hotel: C=75s+1,000; Greenhill Country Club: C=65s+500 Regal Hotel: upper C equals 75 s plus 1,000 ; Greenhill Country Club: upper C equals 65 s plus 500 Regal Hotel: C=1,000s+75; Greenhill Country Club: C=500s+65 Regal Hotel: upper C equals 1,000 s plus 75 ; Greenhill Country Club: upper C equals 500 s plus 65 Regal Hotel: C=75s+500; Greenhill Country Club: C=65s+1,000 Regal Hotel: upper C equals 75 s plus 500 ; Greenhill Country Club: upper C equals 65 s plus 1,000 Regal Hotel: C=500s+75; Greenhill Country Club: C=1,000s+65
Answered by GPT-4o mini
To solve the system of equations:

1. \(-3x - 4y = -23\)
2. \(3x - 5y = 5\)

We can use the method of substitution or elimination. Here, we will use elimination.

First, we can multiply the first equation by \(-1\) to make the coefficients of \(x\) in both equations match:

\[
3x + 4y = 23 \quad \text{(1')}
\]

Now we have:

1. \(3x + 4y = 23\)
2. \(3x - 5y = 5\)

Next, we'll subtract the second equation from the first:

\[
(3x + 4y) - (3x - 5y) = 23 - 5
\]
\[
4y + 5y = 18
\]
\[
9y = 18
\]
\[
y = 2
\]

Now that we have \(y\), we can substitute it back into one of the original equations to solve for \(x\). We'll use the second equation:

\[
3x - 5(2) = 5
\]
\[
3x - 10 = 5
\]
\[
3x = 15
\]
\[
x = 5
\]

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

Thus, the answer is \((5, 2)\).
Answered by GPT-4o mini
To find out how many solutions the system of equations has, let's write down the equations clearly:

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

Now, let's analyze the two equations.

First, we can subtract Equation 1 from Equation 2:

\[
(8x + 6y) - (8x - 6y) = 48 - 48
\]

This simplifies to:

\[
6y + 6y = 0
\]
\[
12y = 0
\]
\[
y = 0
\]

Now that we have \(y = 0\), we can substitute this value back into either of the original equations to solve for \(x\). We'll use Equation 1:

\[
8x - 6(0) = 48
\]
\[
8x = 48
\]
\[
x = 6
\]

So, we found a single solution: \((x, y) = (6, 0)\).

Next, let's examine the nature of the system.

Both equations can be rearranged as follows:

1. \(y = \frac{4}{3}x - 8\) (from Equation 1)
2. \(y = -\frac{4}{3}x + 8\) (from Equation 2)

The first equation has a positive slope, while the second has a negative slope. Thus, they are lines that intersect at one point.

Therefore, the system of equations has exactly **one solution**.

The answer is: **one**.
Answered by GPT-4o mini
To solve the system of equations using the elimination method, we start with the following two equations:

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

**Step 1: Make the coefficients of \(x\) or \(y\) the same.**

To eliminate \(x\), we can multiply Equation 2 by \(2\) so that the coefficients of \(x\) will match:

\[
2(-2x + 4y) = 2(30)
\]
This gives us:
\[
-4x + 8y = 60 \quad \text{(Equation 3)}
\]

Now the system of equations is:

1. \(4x - 6y = -48\) (Equation 1)
2. \(-4x + 8y = 60\) (Equation 3)

**Step 2: Add Equations 1 and 3.**

Now we can add Equation 1 and Equation 3:

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

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

**Step 3: Substitute \(y\) back into one of the original equations to find \(x\).**

We'll use Equation 2 for substitution:

\[
-2x + 4(6) = 30
\]
\[
-2x + 24 = 30
\]
\[
-2x = 30 - 24
\]
\[
-2x = 6
\]
\[
x = -3
\]

**Step 4: Write the solution as an ordered pair.**

The solution to the system of equations is \((x, y) = (-3, 6)\).

Thus, the answer is: \((-3, 6)\).
Answered by GPT-4o mini
To solve the system of equations:

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

Let's start by simplifying Equation 2:

Divide the entire Equation 2 by 5:

\[
x - y = 11 \quad \text{(Equation 3)}
\]

Now we can express \(x\) in terms of \(y\) from Equation 3:

\[
x = y + 11
\]

**Step 1: Substitute \(x\) in Equation 1.**

Now we can substitute \(x\) back into Equation 1:

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

Expanding this gives:

\[
3y + 33 - 4y = 40
\]
\[
-y + 33 = 40
\]

**Step 2: Solve for \(y\)**:

\[
-y = 40 - 33
\]
\[
-y = 7
\]
\[
y = -7
\]

**Step 3: Substitute \(y\) back to find \(x\)**:

Now substitute \(y = -7\) back into Equation 3:

\[
x = -7 + 11
\]
\[
x = 4
\]

**Step 4: Write the solution as an ordered pair**:

The solution to the system of equations is \((x, y) = (4, -7)\).

Thus, the answer is: \((4, -7)\).
Answered by GPT-4o mini
To determine how many solutions the given system of equations has, we can analyze the equations:

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

**Step 1: Simplify the equations.**

Let's see if we can relate these two equations. Notice that we can multiply Equation 2 by \(3\) to see if it helps us compare directly with Equation 1:

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

This gives us:

\[
12x - 15y = 18 \quad \text{(This is exactly Equation 1)}
\]

**Step 2: Analyze the relationship.**

Since multiplying Equation 2 by \(3\) gives us exactly Equation 1, both equations represent the same line. Therefore, every point on the line \(12x - 15y = 18\) is also a solution to \(4x - 5y = 6\).

**Conclusion:**

Since the two equations represent the same line, the system has **infinitely many solutions**.

The answer is: **infinitely many solutions**.
Answered by GPT-4o mini
To create the pair of linear equations representing the cost for each location based on the number of students, we can analyze the information provided for both venues.

### Regal Hotel:
- Fixed ballroom rental fee: \(500\)
- Cost per student for the meal: \(75\)

The equation for the total cost \(C\) at the Regal Hotel as a function of the number of students \(s\) is:

\[
C = 75s + 500
\]

### Greenhill Country Club:
- Fixed ballroom rental fee: \(1,000\)
- Cost per student for the meal: \(65\)

The equation for the total cost \(C\) at the Greenhill Country Club as a function of the number of students \(s\) is:

\[
C = 65s + 1,000
\]

### Summary of the Equations:
- Regal Hotel: \(C = 75s + 500\)
- Greenhill Country Club: \(C = 65s + 1,000\)

So, the correct response representing these two equations is:

**Regal Hotel: \(C = 75s + 500\); Greenhill Country Club: \(C = 65s + 1,000\)**.