6. if ann buys 6 apples, 5 bananas, and 2 pears, she will pay $4.15. if she buys 3 apples, 7 bananas, and 4 pears, she will pay $4.10. if apples are $0.11 less expensive than twice the cost of pears, what is the cost of each item?
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7. a high school’s enrollment is 500 students, which includes freshmen, juniors, and seniors. twice the freshmen enrollment is three times the senior enrollment. the total number of juniors and seniors enrolled is 200 more than the number of freshment enrolled. how many seniors are enrolled.
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8. in the triangle cat. the measure of angel a is three times the measure of angle t. the measure of angel c is twice the sum of the measures of a and t. remember, the sum of the angles of a triangle adds up to 180 degrees. find the measures of the angles.
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*special cases where the calculation will give you an error, so solve by hand to determine if it’s no solution or infinite many.
9. y=-1/2 x + 2
x + 2y = 4
10. x - y = 7
x - y = -4

Question

Accepted Answer

### 6.
# Answer:
Let the cost of an apple be $a$, the cost of a banana be $b$, and the cost of a pear be $p$.
We have the following system of equations:
$$
\begin{cases}
6a + 5b+2p=4.15\
3a + 7b + 4p=4.10\
a = 2p- 0.11
\end{cases}
$$
Substitute $a = 2p-0.11$ into the first two equations:
First equation: $6(2p - 0.11)+5b + 2p=4.15$, which simplifies to $12p-0.66 + 5b+2p=4.15$, or $5b+14p=4.81$
Second - equation: $3(2p - 0.11)+7b + 4p=4.10$, which simplifies to $6p-0.33+7b + 4p=4.10$, or $7b + 10p=4.43$

Multiply the first new - equation by 7 and the second new - equation by 5:
$35b+98p = 33.67$ and $35b+50p=22.15$

Subtract the second new - new equation from the first new - new equation:
$(35b + 98p)-(35b + 50p)=33.67 - 22.15$
$48p=11.52$
$p = 0.24$

Substitute $p = 0.24$ into $a = 2p-0.11$, we get $a=2\times0.24 - 0.11=0.37$

Substitute $a = 0.37$ and $p = 0.24$ into the first original equation $6\times0.37+5b + 2\times0.24=4.15$
$2.22+5b+0.48 = 4.15$
$5b=1.45$
$b = 0.29$

So, apples cost $0.37$ dollars each, bananas cost $0.29$ dollars each, and pears cost $0.24$ dollars each.

# Explanation:
## Step1: Set up equations
Based on the problem description.
## Step2: Substitute $a$
Replace $a$ in the first two equations.
## Step3: Simplify new equations
Combine like - terms.
## Step4: Eliminate $b$
Multiply equations to make $b$ coefficients the same and subtract.
## Step5: Solve for $p$
Find the value of $p$ from the resulting equation.
## Step6: Solve for $a$
Substitute $p$ into the equation for $a$.
## Step7: Solve for $b$
Substitute $a$ and $p$ into the first original equation.

### 7.
# Answer:
Let the number of freshmen be $f$, the number of juniors be $j$, and the number of seniors be $s$.
We know that $f + j + s=500$, $2f = 3s$, and $j + s=f + 200$

From $j + s=f + 200$, we can rewrite it as $j=f + 200 - s$

Substitute $j=f + 200 - s$ into $f + j + s=500$:
$f+(f + 200 - s)+s=500$
$2f+200 = 500$
$2f=300$
$f = 150$

Since $2f = 3s$, substitute $f = 150$ into it:
$2\times150=3s$
$300 = 3s$
$s = 100$

So, there are 100 seniors enrolled.

# Explanation:
## Step1: Set up equations
Based on enrollment relationships.
## Step2: Rewrite $j$ equation
Express $j$ in terms of $f$ and $s$.
## Step3: Substitute $j$
Put $j$ into the total - enrollment equation.
## Step4: Solve for $f$
Find the value of $f$.
## Step5: Solve for $s$
Substitute $f$ into the $f$ - $s$ relationship equation.

### 8.
# Answer:
Let the measure of angle $T$ be $x$ degrees, the measure of angle $A$ be $y$ degrees, and the measure of angle $C$ be $z$ degrees.
We have the following equations:
$y = 3x$ (since the measure of angle $A$ is three times the measure of angle $T$)
$z=2(y + x)$ (since the measure of angle $C$ is twice the sum of the measures of $A$ and $T$)
$x + y+z = 180$ (since the sum of the angles of a triangle is 180 degrees)

Substitute $y = 3x$ into $z=2(y + x)$:
$z=2(3x + x)=8x$

Substitute $y = 3x$ and $z = 8x$ into $x + y+z = 180$:
$x+3x + 8x=180$
$12x=180$
$x = 15$

$y = 3x=45$
$z = 8x = 120$

So, angle $T$ is 15 degrees, angle $A$ is 45 degrees, and angle $C$ is 120 degrees.

# Explanation:
## Step1: Set up equations
Based on angle relationships in the triangle.
## Step2: Substitute $y$ in $z$ equation
Express $z$ in terms of $x$.
## Step3: Substitute $y$ and $z$ in sum - of - angles equation
Get an equation only in terms of $x$.
## Step4: Solve for $x$
Find the value of $x$.
## Step5: Solve for $y$
Substitute $x$ into the $y$ equation.
## Step6: Solve for $z$
Substitute $x$ into the $z$ equation.

### 9.
# Answer:
We have the system of equations $\begin{cases}y=-\frac{1}{2}x + 2\x + 2y=4\end{cases}$
Substitute $y=-\frac{1}{2}x + 2$ into $x + 2y=4$:
$x+2(-\frac{1}{2}x + 2)=4$
$x - x+4 = 4$
$4 = 4$

This system has infinitely many solutions because the two equations represent the same line.

# Explanation:
## Step1: Substitute $y$
Replace $y$ in the second equation.
## Step2: Simplify
Expand and combine like - terms.
## Step3: Analyze result
Since $4 = 4$, the equations are dependent.

### 10.
# Answer:
We have the system of equations $\begin{cases}x - y=7\x - y=-4\end{cases}$
Subtract the second equation from the first equation:
$(x - y)-(x - y)=7-(-4)$
$0 = 11$

This is a contradiction, so the system has no solution.

# Explanation:
## Step1: Subtract equations
To eliminate variables.
## Step2: Analyze result
Since $0
eq11$, it's a contradiction.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

6.

Step1: Set up equations

Step2: Substitute $a$

Step3: Simplify new equations

Step4: Eliminate $b$

Step5: Solve for $p$

Step6: Solve for $a$

Step7: Solve for $b$

7.

Step1: Set up equations

Step2: Rewrite $j$ equation

Step3: Substitute $j$

Step4: Solve for $f$

Step5: Solve for $s$

8.

Step1: Set up equations

Step2: Substitute $y$ in $z$ equation

Step3: Substitute $y$ and $z$ in sum - of - angles equation

Step4: Solve for $x$

Step5: Solve for $y$

Step6: Solve for $z$

9.

Answer: