Question

solve the equations.

1. ( 5(b + 8)+6.9 = 6b - 8 ) 2) ( 4w + 22 = 6w + 3 )

the function below shows the cost to attend the fair if you ride r rides.
( f(r)=5 + 1.75r )

1. what is the y - intercept, and what does it mean?
2. what is the slope, and what does it mean?

create and solve multi - step equations given a context.

1. aiden spent half of her weekly allowance at the movies. to earn more money his parents let him clean the windows in the house for $7. what is his weekly allowance if he ended with $12?
2. the cooking club made some pies to sell at a basketball game to raise money for the new math books. the cafeteria contributed three pies to the sale. each pie was then cut into five pieces and sold. there was a total of 40 pieces to sell. how many pies did the club make?

rewrite the following equations in standard form.

1. ( y=-3x - 3 ) 8) ( y=-10x - 3 )

sketch the graph of the following lines.

1. ( -3x + 2y = 6 ) 10) ( y=-\frac{1}{3}x + 3 )

Explanation:

Step1: Expand the left side of the equation

$$\begin{align*} 5(b + 8)+6.9&=6b-8\\ 5b+40 + 6.9&=6b-8\\ 5b+46.9&=6b-8 \end{align*}$$

Step2: Move the terms with \(b\) to one side

Subtract \(5b\) from both sides:

$$\begin{align*} 5b+46.9-5b&=6b-8-5b\\ 46.9&=b - 8 \end{align*}$$

Step3: Solve for \(b\)

Add \(8\) to both sides:

$$\begin{align*} 46.9+8&=b-8 + 8\\ b&=54.9 \end{align*}$$

Step4: Solve the second equation \(4w + 22=6w+3\)

Move the terms with \(w\) to one side:
Subtract \(4w\) from both sides:

$$\begin{align*} 4w+22-4w&=6w + 3-4w\\ 22&=2w+3 \end{align*}$$

Subtract \(3\) from both sides:

$$\begin{align*} 22-3&=2w+3-3\\ 19&=2w \end{align*}$$

Divide both sides by \(2\):
$$w=\frac{19}{2}=9.5$$

Step5: Analyze the function \(f(r)=5 + 1.75r\) (in the form \(y=mx + b\))

For the \(y\) - intercept (\(r = 0\)):
When \(r = 0\), \(f(0)=5+1.75\times0 = 5\). It means the cost to attend the fair with no rides is \(\$5\) (the initial or base cost).
For the slope \(m = 1.75\). In the context of the function \(f(r)\), the slope represents the cost per ride. So, for each additional ride, the cost to attend the fair increases by \(\$1.75\)

Step6: Solve the problem about Aiden's allowance

Let \(x\) be Aiden's weekly allowance.
He spent \(\frac{x}{2}\) and then earned \(\$7\) and ended with \(\$12\). The equation is \(\frac{x}{2}+7 = 12\)
Subtract \(7\) from both sides: \(\frac{x}{2}=12 - 7=5\)
Multiply both sides by \(2\): \(x = 10\)

Step7: Solve the problem about the Cooking Club's pies

Let \(p\) be the number of pies the club made. The cafeteria contributed \(3\) pies. Each pie is cut into \(5\) pieces.
The equation is \(5(p + 3)=40\)
Divide both sides by \(5\): \(p+3=\frac{40}{5}=8\)
Subtract \(3\) from both sides: \(p=8 - 3=5\)

Step8: Rewrite \(y=-3x - 3\) in standard form (\(Ax+By=C\))

Add \(3x\) to both sides: \(3x+y=- 3\)

Step9: Rewrite \(y=-10x - 3\) in standard form (\(Ax + By=C\))

Add \(10x\) to both sides: \(10x+y=-3\)

Answer:

1. \(b = 54.9\)
2. \(w=9.5\)
3. \(y\) - intercept is \(5\), it means the base cost to attend the fair (cost with \(0\) rides)
4. Slope is \(1.75\), it means the cost per ride
5. \(\$10\)
6. \(5\) pies
7. \(3x + y=-3\)
8. \(10x + y=-3\)