The value of x in this system of equations is 1.

To find the value of \( y \) given that \( x = 1 \), we can substitute \( x \) into one of the equations. We'll use the second equation:

\[ y = -4x + 10 \]

Substituting \( x = 1 \):

\[ y = -4(1) + 10 \] \[ y = -4 + 10 \] \[ y = 6 \]

Thus, the value of \( y \) is:

\[ y = 6 \]

Vineet solved a system of equations by substitution. In his work, he substituted an expression for one of the variables and solved for the value of the other. This resulted in the equation 7 = 9. What can Vineet conclude?

The solution to the system is (7, 9).
The system of equations does not have a solution.
The work must be incorrect because it is not possible to get this equation.
The system of equations has infinitely many solutions.

When Vineet arrived at the equation \( 7 = 9 \), he encountered a contradiction. This type of result indicates that there is no possible solution that satisfies both equations in the system.

Here's the reasoning:

1. Contradictory Result: The equation \( 7 = 9 \) is clearly false and cannot be true.
2. Interpretation: A contradictory result in a system of equations means that the lines represented by the equations are parallel and will never intersect; therefore, there are no points that satisfy both equations simultaneously.

Given these points, Vineet can conclude that:

The system of equations does not have a solution.

A coordinate grid showing Number of T-Shirts on the horizontal x-axis and Total Cost in dollars on the vertical y-axis with 2 lines. The first line is labeled Shirt Box and posses through (0, 30) and the point at (4, 60). The second line is labeled Just Tees and passes through (0, 0) and the point at (6, 63).
The volleyball team at West View High School is comparing T-shirt companies where they can purchase their practice shirts. The graph represents the two companies’ prices. What is the linear equation that represents each T-shirt company?

Shirt Box:

Just Tees:

To determine the linear equations for each T-shirt company based on the given points, we'll use the slope-intercept form of the equation of a line, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Shirt Box:

1. Identify the points: The line passes through the points \( (0, 30) \) and \( (4, 60) \).
2. Calculate the slope (m): \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{60 - 30}{4 - 0} = \frac{30}{4} = 7.5 \]
3. Identify the y-intercept (b): The y-intercept is given directly by the point \( (0, 30) \), so \( b = 30 \).
4. Write the equation: \[ y = 7.5x + 30 \]

Just Tees:

1. Identify the points: The line passes through the points \( (0, 0) \) and \( (6, 63) \).
2. Calculate the slope (m): \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{63 - 0}{6 - 0} = \frac{63}{6} = 10.5 \]
3. Identify the y-intercept (b): The y-intercept is given directly by the point \( (0, 0) \), so \( b = 0 \).
4. Write the equation: \[ y = 10.5x + 0 \quad \text{or simply }\quad y = 10.5x \]

Final Equations:

• Shirt Box: \( y = 7.5x + 30 \)
• Just Tees: \( y = 10.5x \)

The volleyball team at West View High School is comparing T-shirt companies where they can purchase their practice shirts. The two companies, Shirt Box and Just Tees, are represented by this system of equations where x is the number of T-shirts and y is the total cost of the T-shirts.

y = 10.5x

y = 7.5x + 30

How many T-shirts would the volleyball team need to purchase from each company for the total cost to be equal?

For the total cost to be the same for both companies, the volleyball team would need to purchase___
T-shirts from each company for a total of $___
for each company.

To find how many T-shirts the volleyball team needs to purchase from each company for the total cost to be equal, we can set the two equations equal to each other:

\[ 10.5x = 7.5x + 30 \]

Now, we can solve for \( x \):

1. Subtract \( 7.5x \) from both sides: \[ 10.5x - 7.5x = 30 \] \[ 3x = 30 \]

2. Divide both sides by 3: \[ x = 10 \]

So, the volleyball team would need to purchase 10 T-shirts from each company.

Next, we can calculate the total cost for each company when \( x = 10 \):

1. For Shirt Box: \[ y = 7.5x + 30 \] \[ y = 7.5(10) + 30 \] \[ y = 75 + 30 = 105 \]

2. For Just Tees: \[ y = 10.5x \] \[ y = 10.5(10) = 105 \]

Thus, the total cost for each company would be $105.

Summary:

The volleyball team would need to purchase 10 T-shirts from each company for a total of $105 for each company.

A 2 column table with 5 rows. The first column, x, has the entries, negative 4, negative 3, negative 2, negative 1. The second column, y, has the entries 11.5, 17.25, 23, 28.75. A 2 column table with 5 rows. The first column, x, has the entries, negative 4, negative 3, negative 2, negative 1. The second column, y, has the entries, negative 5, negative 7.5, negative 10, negative 12.5.
The tables represent two linear functions. The equation represented by the first table is given below.

y = 5.75x + 34.5

What linear equation is represented by the second table?

What is the solution to the system of equations?

To find the linear equation represented by the second table, we can use the two points given in the table. Let's consider the points from the second table:

For the second table, the points are:

• \((-4, -5)\)
• \((-3, -7.5)\)

Step 1: Calculate the slope (m)

Using the formula for the slope \( m = \frac{y_2 - y_1}{x_2 - x_1} \):

Let’s choose the points \( (-4, -5) \) and \( (-3, -7.5) \):

• \( x_1 = -4, y_1 = -5 \)
• \( x_2 = -3, y_2 = -7.5 \)

Now, calculate the slope:

\[ m = \frac{-7.5 - (-5)}{-3 - (-4)} = \frac{-7.5 + 5}{-3 + 4} = \frac{-2.5}{1} = -2.5 \]

Step 2: Use the slope and one point to write the equation

We will use the point-slope form of the equation \( y - y_1 = m(x - x_1) \) with one of the points. Using point \((-4, -5)\):

\[ y - (-5) = -2.5(x - (-4)) \]

This simplifies to:

\[ y + 5 = -2.5(x + 4) \]

Distributing the slope:

\[ y + 5 = -2.5x - 10 \]

Now, subtract 5 from both sides:

\[ y = -2.5x - 15 \]

The linear equation represented by the second table is:

\[ y = -2.5x - 15 \]

Step 3: Find the solution to the system of equations

Now we need to find the solution to the system of equations:

1. \( y = 5.75x + 34.5 \)
2. \( y = -2.5x - 15 \)

We can set these two equations equal to each other because both are equal to \( y \):

\[ 5.75x + 34.5 = -2.5x - 15 \]

Now, let's solve for \( x \):

1. Add \( 2.5x \) to both sides: \[ 5.75x + 2.5x + 34.5 = -15 \] \[ 8.25x + 34.5 = -15 \]

2. Subtract 34.5 from both sides: \[ 8.25x = -15 - 34.5 \] \[ 8.25x = -49.5 \]

3. Divide by 8.25: \[ x = \frac{-49.5}{8.25} = -6 \]

Step 4: Substitute \( x = -6 \) to find \( y \)

Now we can substitute \( x = -6 \) back into either equation to find \( y \). Let's use the first equation:

\[ y = 5.75(-6) + 34.5 \] \[ y = -34.5 + 34.5 \] \[ y = 0 \]

Final Answers:

The linear equation represented by the second table is:

\[ y = -2.5x - 15 \]

The solution to the system of equations is:

\[ (-6, 0) \]

What linear equation is represented by the second table?

✔ y = –2.5x – 15

What is the solution to the system of equations?

✔ (–6, 0)

Use the system of equations to answer the questions.

2x + 3y = 3

y = 8 – 3x

The value of y from the second equation is substituted back into the first equation. What is the resulting equation?

What is the value of x?

What is the value of y?

Given the system of equations:

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

Step 1: Substitute \( y \) from the second equation into the first equation

Substituting \( y \) from the second equation into the first equation:

\[ 2x + 3(8 - 3x) = 3 \]

Step 2: Simplify the equation

Now, distribute the \( 3 \) in the equation:

\[ 2x + 24 - 9x = 3 \]

Combine like terms:

\[ 2x - 9x + 24 = 3 \] \[ -7x + 24 = 3 \]

Step 3: Solve for \( x \)

Now we can isolate \( x \):

1. Subtract \( 24 \) from both sides:

\[ -7x = 3 - 24 \] \[ -7x = -21 \]

2. Divide by \( -7 \):

\[ x = \frac{-21}{-7} = 3 \]

Step 4: Solve for \( y \)

Now that we have \( x \), we can find \( y \) by substituting \( x = 3 \) back into the second equation:

\[ y = 8 - 3(3) \] \[ y = 8 - 9 \] \[ y = -1 \]

Summary of Results:

• The resulting equation after substituting is \( 2x + 3(8 - 3x) = 3 \).
• The value of \( x \) is \( 3 \).
• The value of \( y \) is \( -1 \).