Question

reteach to build understanding 5-2 what is the solution of the system? graph the lines. the point of intersection is (3, 8). y = 2x + 2 8 ? 2(3) + 2 8 ? 6 + 2 8 = 8 √ y = 4x - 4 8 ? 4(3) - 4 8 ? 12 - 4 8 = 8 √ the solution is x = 3 and y = 8. y = 2x + 2 y = 4x - 4 (graph of the two lines with intersection at (3,8)) a rental car company charges a one time fee of $50 plus $1 per mile. another rental car company charges a one time fee of $10 plus $2 per mile. the equations at right represent the total cost, t, for renting a car for m miles at each company. for how many miles is the cost the same at both companies? t = m + 50 t = 2m + 10 1. graph the lines represented by the equations. 2. what is the point of intersection? 3. substitute the coordinates of the intersection point in each equation. do the coordinates represent a solution to both equations? t = m + 50 □ ? □ + 50 □ = □ t = 2m + 10 □ ? 2(□) + 10 □ ? □ + 10 □ = □ (rental car costs graph with t (cost $) and m (miles)) 4. for how many miles is the cost the same at both companies? on the back! 5. manny has 85 cents in nickels and dimes. he has 10 coins altogether. the equations at right relate the number of nickels, x, and the number of dimes, y. graph each equation. how many nickels and how many dimes does manny have? x + y = 10 5x + 10y = 85

Explanation:

Sub - question 1: Graph the lines represented by the equations.

To graph \(t = m + 50\) and \(t=2m + 10\):

• For \(t=m + 50\):
• When \(m = 0\), \(t=0 + 50=50\), so the y - intercept (in terms of \(t\) and \(m\)) is \((0,50)\).
• When \(t = 0\), \(0=m + 50\), then \(m=- 50\) (but since miles can't be negative, we focus on non - negative \(m\) values). The slope is \(1\) (since the equation is in the form \(t=1\times m+50\)).
• For \(t = 2m+10\):
• When \(m = 0\), \(t=0 + 10 = 10\), so the y - intercept is \((0,10)\).
• When \(t = 0\), \(0=2m + 10\), then \(2m=-10\) and \(m=- 5\) (again, we focus on non - negative \(m\) for the context of miles). The slope is \(2\).

We can plot points for non - negative \(m\) values. For example, for \(t=m + 50\), when \(m = 10\), \(t=60\); when \(m = 20\), \(t = 70\). For \(t=2m + 10\), when \(m = 10\), \(t=30\); when \(m = 20\), \(t=50\). Then draw the two lines on the coordinate plane with \(m\) on the x - axis and \(t\) on the y - axis.

Sub - question 2: What is the point of intersection?

To find the point of intersection, we set the two equations equal to each other since at the intersection point, \(t\) (and \(m\)) have the same values for both equations.
Set \(m + 50=2m + 10\).

Step 1: Solve for \(m\)

Subtract \(m\) from both sides: \(m+50 - m=2m + 10 - m\), which simplifies to \(50=m + 10\).
Then subtract \(10\) from both sides: \(50-10=m+10 - 10\), so \(m = 40\).

Step 2: Solve for \(t\)

Substitute \(m = 40\) into \(t=m + 50\) (we could also use \(t = 2m+10\)).
\(t=40 + 50=90\).
So the point of intersection is \((40,90)\) (where \(m = 40\) and \(t = 90\)).

Sub - question 3: Substitute the coordinates of the intersection point in each equation.

For the equation \(t=m + 50\):
Substitute \(m = 40\) and \(t = 90\) into \(t=m + 50\).
We get \(90\stackrel{?}{=}40 + 50\), and \(90=90\), which is true.
For the equation \(t=2m + 10\):
Substitute \(m = 40\) and \(t = 90\) into \(t=2m + 10\).
We get \(90\stackrel{?}{=}2\times40+10\).
First, calculate \(2\times40 = 80\), then \(80 + 10=90\). So \(90=90\), which is true.
So the coordinates \((m = 40,t = 90)\) represent a solution to both equations.
Filling in the blanks:
For \(t=m + 50\): \(\boldsymbol{90}\stackrel{?}{=}\boldsymbol{40}+50\), \(\boldsymbol{90}=\boldsymbol{90}\)
For \(t = 2m+10\): \(\boldsymbol{90}\stackrel{?}{=}2(\boldsymbol{40})+10\), \(\boldsymbol{90}\stackrel{?}{=}\boldsymbol{80}+10\), \(\boldsymbol{90}=\boldsymbol{90}\)

Sub - question 4: For how many miles is the cost the same at both companies?

From the point of intersection, we found that \(m = 40\). So the cost is the same when the number of miles \(m = 40\).

Sub - question 5: How many nickels and dimes does Manny have?

We have the system of equations:
\(

$$\begin{cases}x + y=10\\5x + 10y=85\end{cases}$$

\)
From the first equation \(x=10 - y\).

Step 1: Substitute \(x = 10 - y\) into the second equation

Substitute \(x\) into \(5x + 10y=85\):
\(5(10 - y)+10y=85\)

Step 2: Expand and simplify

Expand \(5(10 - y)\): \(50-5y+10y=85\)
Combine like terms: \(50 + 5y=85\)

Step 3: Solve for \(y\)

Subtract \(50\) from both sides: \(5y=85 - 50=35\)
Divide both sides by \(5\): \(y = 7\)

Step 4: Solve for \(x\)

Substitute \(y = 7\) into \(x = 10 - y\): \(x=10 - 7 = 3\)
So Manny has 3 nickels and 7 dimes.

Answer:

s:

1. (Graphing steps as above)
2. The point of intersection is \((40,90)\) (where \(m = 40\) and \(t = 90\))
3. For \(t=m + 50\): \(90\stackrel{?}{=}40 + 50\), \(90 = 90\); For \(t=2m + 10\): \(90\stackrel{?}{=}2(40)+10\), \(90\stackrel{?}{=}80 + 10\), \(90 = 90\) (the coordinates represent a solution to both equations)
4. \(40\) miles
5. Manny has 3 nickels and 7 dimes.