the equation is y = -1.24x + 97.40

Question

a) What is the predicted heating cost for a month with an average temperature of 0?

(b) What is the predicted heating cost for a month with an average temperature of 25?

(c) For an increase of one degree Fahrenheit, what is the predicted decrease in the monthly heating cost?

Answer 1

To solve the questions based on the equation \( y = -1.24x + 97.40 \), where \( y \) represents the predicted heating cost and \( x \) represents the average temperature, we can substitute the values for \( x \) to find \( y \).

a) Predicted heating cost for a month with an average temperature of 0:

Substituting \( x = 0 \) into the equation:

\[ y = -1.24(0) + 97.40 = 97.40 \]

The predicted heating cost for a month with an average temperature of 0 degrees Fahrenheit is $97.40.

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b) Predicted heating cost for a month with an average temperature of 25:

Substituting \( x = 25 \) into the equation:

\[ y = -1.24(25) + 97.40 \]

Calculating further:

\[ y = -31 + 97.40 = 66.40 \]

The predicted heating cost for a month with an average temperature of 25 degrees Fahrenheit is $66.40.

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c) For an increase of one degree Fahrenheit, what is the predicted decrease in the monthly heating cost?

From the equation \( y = -1.24x + 97.40 \), the coefficient of \( x \) is \(-1.24\). This means that for each increase of 1 degree Fahrenheit in temperature (\( x \)), the heating cost (\( y \)) decreases by 1.24.

The predicted decrease in the monthly heating cost for an increase of one degree Fahrenheit is $1.24.

Answer 2

For each ordered pair, determine whether it is a solution to the system of equations.
3x - 2y = -4
2x + 5y = -9
Is it a solution?
Yes No
(-2, -1)
(8, -5)
(4, 8)
(0, 3)

Answer 3

To determine whether each ordered pair is a solution to the system of equations:

1. \(3x - 2y = -4\)
2. \(2x + 5y = -9\)

we will substitute the values of \(x\) and \(y\) from each ordered pair into both equations.

1. For the point \((-2, -1)\):

• Substitute \(x = -2\) and \(y = -1\) into the first equation: \[ 3(-2) - 2(-1) = -4 \implies -6 + 2 = -4 \implies -4 = -4 \quad \text{(True)} \]
• Now substitute into the second equation: \[ 2(-2) + 5(-1) = -9 \implies -4 - 5 = -9 \implies -9 = -9 \quad \text{(True)} \] So, \((-2, -1)\) is a solution. Yes.

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2. For the point \((8, -5)\):

• Substitute \(x = 8\) and \(y = -5\) into the first equation: \[ 3(8) - 2(-5) = -4 \implies 24 + 10 = -4 \implies 34 \neq -4 \quad \text{(False)} \] So, \((8, -5)\) is not a solution. No.

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3. For the point \((4, 8)\):

• Substitute \(x = 4\) and \(y = 8\) into the first equation: \[ 3(4) - 2(8) = -4 \implies 12 - 16 = -4 \implies -4 = -4 \quad \text{(True)} \]
• Now substitute into the second equation: \[ 2(4) + 5(8) = -9 \implies 8 + 40 = -9 \implies 48 \neq -9 \quad \text{(False)} \] So, \((4, 8)\) is not a solution. No.

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4. For the point \((0, 3)\):

• Substitute \(x = 0\) and \(y = 3\) into the first equation: \[ 3(0) - 2(3) = -4 \implies 0 - 6 = -4 \implies -6 \neq -4 \quad \text{(False)} \] So, \((0, 3)\) is not a solution. No.

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Summary of results:

• \((-2, -1)\): Yes
• \((8, -5)\): No
• \((4, 8)\): No
• \((0, 3)\): No

Answer 4

make an equation from the plotted points.
(0, 5)
(4,4)
(0,-4)

Answer 5

To create an equation from the given points \((0, 5)\), \((4, 4)\), and \((0, -4)\), we first need to identify if these points can represent a linear relationship.

Step 1: Identify the points moving along the x and y axes

1. The points \((0, 5)\) and \((0, -4)\) have the same x-coordinate, which means they both lie on the y-axis at \(x = 0\). This may indicate that we could represent two separate segments of a piecewise function if needed.

2. The point \((4, 4)\) has a unique x-value.

Step 2: Check if the points can form a linear equation

Let's first examine the point \((4, 4)\) and the y-intercept points, \((0, 5)\) and \((0, -4)\).

Step 3: Determine if a linear relationship exists

We can attempt to connect the points \((4, 4)\) to the intercepts.

Create the equation using the point-slope form:

Using the points \((0, 5)\) and \((4, 4)\):

1. Calculate the slope \(m\):

\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 5}{4 - 0} = \frac{-1}{4} = -0.25 \]

2. Use the point-slope form of the line equation \(y - y_1 = m(x - x_1)\): Using point \((0, 5)\):

\[ y - 5 = -0.25(x - 0) \]

\[ y - 5 = -0.25x \]

\[ y = -0.25x + 5 \]

Step 4: Check the other point \((0, -4)\)

However, to connect with the point \((0, -4)\), we can write the equations of the lines formed by the segments.

1. Identify another line for the segment between points \((0,5)\) and \((0,-4)\).

Creating equation for line through (0, 5) and (0, -4):

Since both points have the same x-coordinate, they are vertical, and the equation will be \[ x = 0 \] with any y value between 5 and -4.

Conclusion

The main linear equation from the points \((0, 5)\) and \((4, 4)\) is: \[ y = -0.25x + 5 \]

If you want to describe the whole system, you may need to express conditions for the segments:

• From \((0, 5)\) down to \((0, -4)\), it's just a portion of the y-axis.
• The linear segment can be described as moving from \((0, 5)\) to \((4, 4)\) and another potentially descending line from \((4, 4)\) to \((0, -4)\).

Alternate line equations can be derived based on parts and the behavior of the vertical stretch through additional segments if required. Let me know if how you want these points expressed!

Answer 6

make two equations that will intersect at these points

Answer 7

To create two equations that will intersect at the points \((0, 5)\), \((4, 4)\), and \((0, -4)\), we can derive one equation that intersects through \((0, 5)\) and \((4, 4)\), and another that intersects through \((0, -4)\) and the other points.

Step 1: Create the First Equation

We'll start with the points \((0, 5)\) and \((4, 4)\).

1. Finding the slope \(m\): \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 5}{4 - 0} = \frac{-1}{4} = -0.25 \]

2. Using point-slope form with \((0, 5)\): \[ y - 5 = -0.25(x - 0) \] Thus, the equation simplifies to: \[ y = -0.25x + 5 \]

Step 2: Create the Second Equation

Next, we need to create a linear equation that connects points \((0, -4)\) and \((4, 4)\).

1. Finding the slope \(m\) between \((0, -4)\) and \((4, 4)\): \[ m = \frac{4 - (-4)}{4 - 0} = \frac{4 + 4}{4} = \frac{8}{4} = 2 \]

2. Using point-slope form with \((0, -4)\): \[ y - (-4) = 2(x - 0) \] This simplifies to: \[ y + 4 = 2x \] Hence, the equation can be written as: \[ y = 2x - 4 \]

Summary of Equations

The two equations that intersect at the points \((0, 5)\), \((4, 4)\), and \((0, -4)\) are:

1. First equation: \[ y = -0.25x + 5 \]

2. Second equation: \[ y = 2x - 4 \]

Points of Intersection

• Intersection at (0, 5): Substituting \(x = 0\):

  • For the first equation: \(y = -0.25(0) + 5 = 5\)
  • For the second equation: \(y = 2(0) - 4 = -4\) (which it does not).

• Intersection at (4, 4): Substituting \(x = 4\):

  • For the first equation: \(y = -0.25(4) + 5 = 4\)
  • For the second equation: \(y = 2(4) - 4 = 4\)

• Intersection at (0, -4): Substituting \(x = 0\):

  • For the first equation: \(y = -0.25(0) + 5 = 5\) (which it does not).
  • For the second equation: \(y = -4\)

Thus, the second equation uses the point that intersects through other distinct trajectories. You can plot these lines to see that they describe the behavior around those intersections. Let me know if further refinements or checks are needed!

Answer 8

use substitution to solve the system
4x - 5y = -13
x = 2y - 4

Answer 9

To solve the system of equations using substitution, we start with the given equations:

1. \(4x - 5y = -13\) \quad (Equation 1)
2. \(x = 2y - 4\) \quad (Equation 2)

Since Equation 2 already expresses \(x\) in terms of \(y\), we can substitute that expression for \(x\) into Equation 1.

Step 1: Substitute Equation 2 into Equation 1

We substitute \(x = 2y - 4\) into Equation 1:

\[ 4(2y - 4) - 5y = -13 \]

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

Expand and simplify the equation:

\[ 8y - 16 - 5y = -13 \]

Combine like terms:

\[ 8y - 5y - 16 = -13 \]

This simplifies to:

\[ 3y - 16 = -13 \]

Now, add 16 to both sides:

\[ 3y = -13 + 16 \]

\[ 3y = 3 \]

Divide both sides by 3:

\[ y = 1 \]

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

Now that we have \(y = 1\), we can substitute this value back into Equation 2 to find \(x\):

\[ x = 2(1) - 4 \]

\[ x = 2 - 4 \]

\[ x = -2 \]

Final Solution

The solution to the system of equations is:

\[ (x, y) = (-2, 1) \]

We can check this solution by substituting \(x = -2\) and \(y = 1\) back into the original equations to verify:

Checking Equation 1:

\[ 4(-2) - 5(1) = -13 \implies -8 - 5 = -13 \quad \text{(True)} \]

Checking Equation 2:

\[ x = 2(1) - 4 \implies -2 = 2 - 4 \quad \text{(True)} \]

Both equations are satisfied, confirming the solution is correct.

Answer: \((x, y) = (-2, 1)\)

Answer 10

3x + y = 3
x + y = 2
solve the system of equations

Answer 11

To solve the system of equations:

1. \(3x + y = 3\) \quad (Equation 1)
2. \(x + y = 2\) \quad (Equation 2)

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

Step 1: Solve Equation 2 for \(y\)

From Equation 2 \(x + y = 2\), we can express \(y\) in terms of \(x\):

\[ y = 2 - x \]

Step 2: Substitute \(y\) into Equation 1

Now we substitute \(y = 2 - x\) into Equation 1:

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

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

Combine the terms:

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

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

\[ 2x + 2 = 3 \]

Subtract 2 from both sides:

\[ 2x = 3 - 2 \]

\[ 2x = 1 \]

Now divide by 2:

\[ x = \frac{1}{2} \]

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

Now that we have \(x\), substitute \(x = \frac{1}{2}\) back into the expression for \(y\):

\[ y = 2 - \frac{1}{2} \]

\[ y = \frac{4}{2} - \frac{1}{2} \]

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

Final Solution

The solution to the system of equations is:

\[ (x, y) = \left(\frac{1}{2}, \frac{3}{2}\right) \]

Verification

To check if this solution is correct, we can substitute \(x = \frac{1}{2}\) and \(y = \frac{3}{2}\) back into both original equations.

Checking Equation 1:

\[ 3\left(\frac{1}{2}\right) + \frac{3}{2} = 3 \]

\[ \frac{3}{2} + \frac{3}{2} = 3 \quad \text{(True)} \]

Checking Equation 2:

\[ \frac{1}{2} + \frac{3}{2} = 2 \]

\[ 2 = 2 \quad \text{(True)} \]

Both equations are satisfied, confirming the solution is correct.

Final Answer: \((x, y) = \left(\frac{1}{2}, \frac{3}{2}\right)\)