Question
Malik collects rare stamps and has a total of 212 stamps. He has 34 more domestic stamps than foreign stamps. Let x represent the number of domestic stamps and let y represent the number of foreign stamps.
Which equation represents the total number of stamps Malik collected?
Which equation represents the difference in the number of foreign and domestic stamps Malik collected?
Which system of linear equations represents the situation?
Which equation represents the total number of stamps Malik collected?
Which equation represents the difference in the number of foreign and domestic stamps Malik collected?
Which system of linear equations represents the situation?
Answers
Answer
Malik collects rare stamps and has a total of 212 stamps. He has 34 more domestic stamps than foreign stamps. Let x represent the number of domestic stamps and let y represent the number of foreign stamps.
Which equation represents the total number of stamps Malik collected?
✔ x + y = 212
Which equation represents the difference in the number of foreign and domestic stamps Malik collected?
✔ x – y = 34
Which system of linear equations represents the situation?
✔ x – y = 34 and x + y = 212
Malik collects rare stamps and has a total of 212 stamps. He has 34 more domestic stamps than foreign stamps. Let x represent the number of domestic stamps and let y represent the number of foreign stamps.
This system of equations models the given information for both stamp types.
x – y = 34
x + y = 212
Solve the system of equations.
How many foreign stamps does Malik have?
foreign stamps
How many domestic stamps does Malik have?
domestic stamps
Which equation represents the total number of stamps Malik collected?
✔ x + y = 212
Which equation represents the difference in the number of foreign and domestic stamps Malik collected?
✔ x – y = 34
Which system of linear equations represents the situation?
✔ x – y = 34 and x + y = 212
Malik collects rare stamps and has a total of 212 stamps. He has 34 more domestic stamps than foreign stamps. Let x represent the number of domestic stamps and let y represent the number of foreign stamps.
This system of equations models the given information for both stamp types.
x – y = 34
x + y = 212
Solve the system of equations.
How many foreign stamps does Malik have?
foreign stamps
How many domestic stamps does Malik have?
domestic stamps
Answer
Solve the system of equations.
2.5y + 3x = 27
5x – 2.5y = 5
What equation is the result of adding the two equations?
What is the solution to the system?
2.5y + 3x = 27
5x – 2.5y = 5
What equation is the result of adding the two equations?
What is the solution to the system?
Answer
Solve the system of equations.
2.5y + 3x = 27
5x – 2.5y = 5
What equation is the result of adding the two equations?
✔ 8x = 32
What is the solution to the system?
✔ (4, 6)
2.5y + 3x = 27
5x – 2.5y = 5
What equation is the result of adding the two equations?
✔ 8x = 32
What is the solution to the system?
✔ (4, 6)
Answer
Solve the system of equations.
2.5y + 3x = 27
5x – 2.5y = 5
What equation is the result of adding the two equations?
✔ 8x = 32
What is the solution to the system?
✔ (4, 6)
Solve the system of equations.
6x – 3y = 3
–2x + 6y = 14
What number would you multiply the second equation by in order to eliminate the x-terms when adding to the first equation?
What number would you multiply the first equation by in order to eliminate the y-terms when adding to the second equation?
2.5y + 3x = 27
5x – 2.5y = 5
What equation is the result of adding the two equations?
✔ 8x = 32
What is the solution to the system?
✔ (4, 6)
Solve the system of equations.
6x – 3y = 3
–2x + 6y = 14
What number would you multiply the second equation by in order to eliminate the x-terms when adding to the first equation?
What number would you multiply the first equation by in order to eliminate the y-terms when adding to the second equation?
Answer
Follow the directions to solve the system of equations by elimination.
8x + 7y = 39
4x – 14y = –68
Multiply the first equation to enable the elimination of the y-term.
Add the equations to eliminate the y-terms.
Solve the new equation for the x-value.
Substitute the x-value back into either original equation to find the y-value.
Check the solution.
The solution to the system of equations is (__
,
__).
8x + 7y = 39
4x – 14y = –68
Multiply the first equation to enable the elimination of the y-term.
Add the equations to eliminate the y-terms.
Solve the new equation for the x-value.
Substitute the x-value back into either original equation to find the y-value.
Check the solution.
The solution to the system of equations is (__
,
__).
Answered by
GPT-4o mini
To solve the problem, we need to define the equations based on the information provided.
Let:
- \( x \) = number of domestic stamps
- \( y \) = number of foreign stamps
1. **Equation representing the total number of stamps:**
Since Malik has a total of 212 stamps, the equation representing this total is:
\[
x + y = 212
\]
2. **Equation representing the difference in the number of foreign and domestic stamps:**
Since Malik has 34 more domestic stamps than foreign stamps, this gives us the equation:
\[
x = y + 34
\]
or equivalently,
\[
x - y = 34
\]
3. **System of linear equations representing the situation:**
We can now combine both equations into a system:
\[
\begin{cases}
x + y = 212 \\
x - y = 34
\end{cases}
\]
This system of equations can be used to find the values of \( x \) and \( y \) (the number of domestic and foreign stamps, respectively).
Let:
- \( x \) = number of domestic stamps
- \( y \) = number of foreign stamps
1. **Equation representing the total number of stamps:**
Since Malik has a total of 212 stamps, the equation representing this total is:
\[
x + y = 212
\]
2. **Equation representing the difference in the number of foreign and domestic stamps:**
Since Malik has 34 more domestic stamps than foreign stamps, this gives us the equation:
\[
x = y + 34
\]
or equivalently,
\[
x - y = 34
\]
3. **System of linear equations representing the situation:**
We can now combine both equations into a system:
\[
\begin{cases}
x + y = 212 \\
x - y = 34
\end{cases}
\]
This system of equations can be used to find the values of \( x \) and \( y \) (the number of domestic and foreign stamps, respectively).
Answered by
GPT-4o mini
To solve the system of equations given by:
\[
\begin{cases}
x - y = 34 \quad \text{(1)} \\
x + y = 212 \quad \text{(2)}
\end{cases}
\]
we can use substitution or elimination. Here, let's use substitution.
1. From equation (1), we can express \( x \) in terms of \( y \):
\[
x = y + 34
\]
2. Now, we can substitute \( x \) in equation (2):
\[
(y + 34) + y = 212
\]
3. Simplify this:
\[
2y + 34 = 212
\]
4. Subtract 34 from both sides:
\[
2y = 178
\]
5. Divide by 2:
\[
y = 89
\]
Now that we have the number of foreign stamps, we can find the number of domestic stamps using equation (1):
6. Substitute \( y \) back into the expression for \( x \):
\[
x = y + 34 = 89 + 34 = 123
\]
### Summary of the results:
- Number of foreign stamps \( y \): **89**
- Number of domestic stamps \( x \): **123**
Thus, Malik has **89 foreign stamps** and **123 domestic stamps**.
\[
\begin{cases}
x - y = 34 \quad \text{(1)} \\
x + y = 212 \quad \text{(2)}
\end{cases}
\]
we can use substitution or elimination. Here, let's use substitution.
1. From equation (1), we can express \( x \) in terms of \( y \):
\[
x = y + 34
\]
2. Now, we can substitute \( x \) in equation (2):
\[
(y + 34) + y = 212
\]
3. Simplify this:
\[
2y + 34 = 212
\]
4. Subtract 34 from both sides:
\[
2y = 178
\]
5. Divide by 2:
\[
y = 89
\]
Now that we have the number of foreign stamps, we can find the number of domestic stamps using equation (1):
6. Substitute \( y \) back into the expression for \( x \):
\[
x = y + 34 = 89 + 34 = 123
\]
### Summary of the results:
- Number of foreign stamps \( y \): **89**
- Number of domestic stamps \( x \): **123**
Thus, Malik has **89 foreign stamps** and **123 domestic stamps**.
Answered by
GPT-4o mini
To solve the system of equations:
\[
\begin{cases}
2.5y + 3x = 27 \quad \text{(1)} \\
5x - 2.5y = 5 \quad \text{(2)}
\end{cases}
\]
### Step 1: Add the Two Equations
First, let's rearrange equation (2) to make it easier to add the equations directly.
We can align the equations for addition:
- From (1): \( 2.5y + 3x = 27 \)
- From (2): \( 5x - 2.5y = 5 \)
Now, we can add the two equations:
\[
(2.5y + 3x) + (5x - 2.5y) = 27 + 5
\]
Combining like terms:
\[
(2.5y - 2.5y) + (3x + 5x) = 32
\]
\[
0 + 8x = 32
\]
\[
8x = 32
\]
### Step 2: Solve for \( x \)
Divide both sides by 8:
\[
x = 4
\]
### Step 3: Substitute \( x \) back into one of the original equations
Now substitute \( x = 4 \) into equation (1) to find \( y \):
\[
2.5y + 3(4) = 27
\]
\[
2.5y + 12 = 27
\]
Subtract 12 from both sides:
\[
2.5y = 15
\]
Divide by 2.5:
\[
y = 6
\]
### Summary of Results
- The result of adding the two equations is: **\( 8x = 32 \)**
- The solution to the system is \( (x, y) = (4, 6) \).
Thus, the solution to the system is:
\[
x = 4, \quad y = 6
\]
\[
\begin{cases}
2.5y + 3x = 27 \quad \text{(1)} \\
5x - 2.5y = 5 \quad \text{(2)}
\end{cases}
\]
### Step 1: Add the Two Equations
First, let's rearrange equation (2) to make it easier to add the equations directly.
We can align the equations for addition:
- From (1): \( 2.5y + 3x = 27 \)
- From (2): \( 5x - 2.5y = 5 \)
Now, we can add the two equations:
\[
(2.5y + 3x) + (5x - 2.5y) = 27 + 5
\]
Combining like terms:
\[
(2.5y - 2.5y) + (3x + 5x) = 32
\]
\[
0 + 8x = 32
\]
\[
8x = 32
\]
### Step 2: Solve for \( x \)
Divide both sides by 8:
\[
x = 4
\]
### Step 3: Substitute \( x \) back into one of the original equations
Now substitute \( x = 4 \) into equation (1) to find \( y \):
\[
2.5y + 3(4) = 27
\]
\[
2.5y + 12 = 27
\]
Subtract 12 from both sides:
\[
2.5y = 15
\]
Divide by 2.5:
\[
y = 6
\]
### Summary of Results
- The result of adding the two equations is: **\( 8x = 32 \)**
- The solution to the system is \( (x, y) = (4, 6) \).
Thus, the solution to the system is:
\[
x = 4, \quad y = 6
\]
Answered by
GPT-4o mini
It looks like you've provided the correct answers for the system of equations.
To summarize:
1. The result of adding the two equations:
\[
8x = 32
\]
2. The solution to the system:
\[
(x, y) = (4, 6)
\]
This indicates that Malik must have **4** as the value of \(x\) and **6** as the value of \(y\).
If you need any further elaboration or assistance on this topic or others, feel free to ask!
To summarize:
1. The result of adding the two equations:
\[
8x = 32
\]
2. The solution to the system:
\[
(x, y) = (4, 6)
\]
This indicates that Malik must have **4** as the value of \(x\) and **6** as the value of \(y\).
If you need any further elaboration or assistance on this topic or others, feel free to ask!
Answered by
GPT-4o mini
Let's solve the system of equations:
\[
\begin{cases}
6x - 3y = 3 \quad \text{(1)} \\
-2x + 6y = 14 \quad \text{(2)}
\end{cases}
\]
### Step 1: To Eliminate the x-terms
To eliminate the \(x\)-terms when adding these equations, we want the coefficients of \(x\) in both equations to be the same (but opposite in sign).
- The first equation has a coefficient of \(6\) for \(x\).
- The second equation has a coefficient of \(-2\) for \(x\).
To make the coefficients equal, we can multiply the second equation by \(3\):
\[
3 \cdot (-2x + 6y) = 3 \cdot 14
\]
This gives:
\[
-6x + 18y = 42 \quad \text{(new equation 2)}
\]
Now our system looks like this:
\[
\begin{cases}
6x - 3y = 3 \\
-6x + 18y = 42
\end{cases}
\]
Adding these will eliminate the \(x\)-terms.
### Answer for the first part:
**You would multiply the second equation by \(3\)**.
### Step 2: To Eliminate the y-terms
To eliminate the \(y\)-terms when adding the equations, we want to make the coefficients of \(y\) in both equations equal.
- The first equation has a coefficient of \(-3\) for \(y\).
- The second equation has a coefficient of \(6\) for \(y\).
To make them equal, we can multiply the first equation by \(2\):
\[
2 \cdot (6x - 3y) = 2 \cdot 3
\]
This gives:
\[
12x - 6y = 6 \quad \text{(new equation 1)}
\]
Now our system looks like this:
\[
\begin{cases}
12x - 6y = 6 \\
-2x + 6y = 14
\end{cases}
\]
Adding these will eliminate the \(y\)-terms.
### Answer for the second part:
**You would multiply the first equation by \(2\)**.
### Summary:
1. **Multiply the second equation by \(3\) to eliminate \(x\) terms**.
2. **Multiply the first equation by \(2\) to eliminate \(y\) terms**.
\[
\begin{cases}
6x - 3y = 3 \quad \text{(1)} \\
-2x + 6y = 14 \quad \text{(2)}
\end{cases}
\]
### Step 1: To Eliminate the x-terms
To eliminate the \(x\)-terms when adding these equations, we want the coefficients of \(x\) in both equations to be the same (but opposite in sign).
- The first equation has a coefficient of \(6\) for \(x\).
- The second equation has a coefficient of \(-2\) for \(x\).
To make the coefficients equal, we can multiply the second equation by \(3\):
\[
3 \cdot (-2x + 6y) = 3 \cdot 14
\]
This gives:
\[
-6x + 18y = 42 \quad \text{(new equation 2)}
\]
Now our system looks like this:
\[
\begin{cases}
6x - 3y = 3 \\
-6x + 18y = 42
\end{cases}
\]
Adding these will eliminate the \(x\)-terms.
### Answer for the first part:
**You would multiply the second equation by \(3\)**.
### Step 2: To Eliminate the y-terms
To eliminate the \(y\)-terms when adding the equations, we want to make the coefficients of \(y\) in both equations equal.
- The first equation has a coefficient of \(-3\) for \(y\).
- The second equation has a coefficient of \(6\) for \(y\).
To make them equal, we can multiply the first equation by \(2\):
\[
2 \cdot (6x - 3y) = 2 \cdot 3
\]
This gives:
\[
12x - 6y = 6 \quad \text{(new equation 1)}
\]
Now our system looks like this:
\[
\begin{cases}
12x - 6y = 6 \\
-2x + 6y = 14
\end{cases}
\]
Adding these will eliminate the \(y\)-terms.
### Answer for the second part:
**You would multiply the first equation by \(2\)**.
### Summary:
1. **Multiply the second equation by \(3\) to eliminate \(x\) terms**.
2. **Multiply the first equation by \(2\) to eliminate \(y\) terms**.
Answered by
GPT-4o mini
Let's solve the system of equations using the elimination method step by step.
Given equations:
\[
\begin{cases}
8x + 7y = 39 \quad \text{(1)} \\
4x - 14y = -68 \quad \text{(2)}
\end{cases}
\]
### Step 1: Multiply the First Equation
To eliminate the \(y\)-term, we can multiply the first equation (1) by \(2\) so that the coefficients of \(y\) in both equations will match in magnitude (since \(7\) in equation (1) becomes \(14\) after multiplication):
\[
2(8x + 7y) = 2(39) \\
16x + 14y = 78 \quad \text{(new equation 1)}
\]
### Step 2: Write the System of Equations
Now we have the modified system of equations:
\[
\begin{cases}
16x + 14y = 78 \quad \text{(new equation 1)} \\
4x - 14y = -68 \quad \text{(2)}
\end{cases}
\]
### Step 3: Add the Equations to Eliminate the \(y\)-terms
Now add the two equations together to eliminate the \(y\)-terms:
\[
(16x + 14y) + (4x - 14y) = 78 + (-68)
\]
This simplifies to:
\[
16x + 4x + 14y - 14y = 78 - 68
\]
\[
20x = 10
\]
### Step 4: Solve the New Equation for \(x\)
Now solve for \(x\):
\[
x = \frac{10}{20} = \frac{1}{2}
\]
### Step 5: Substitute the \(x\)-value into Either Original Equation
Now substitute \(x = \frac{1}{2}\) back into one of the original equations to find \(y\). Let's use equation (1):
\[
8\left(\frac{1}{2}\right) + 7y = 39
\]
This simplifies to:
\[
4 + 7y = 39
\]
Subtract \(4\) from both sides:
\[
7y = 39 - 4
\]
\[
7y = 35
\]
Now divide by \(7\):
\[
y = 5
\]
### Step 6: Check the Solution
To check, substitute \(x = \frac{1}{2}\) and \(y = 5\) into both original equations.
**For equation (1):**
\[
8\left(\frac{1}{2}\right) + 7(5) = 39 \\
4 + 35 = 39 \quad \text{(True)}
\]
**For equation (2):**
\[
4\left(\frac{1}{2}\right) - 14(5) = -68 \\
2 - 70 = -68 \quad \text{(True)}
\]
Both equations hold true with the given values.
### Final Answer
The solution to the system of equations is:
\[
\left(\frac{1}{2}, 5\right)
\]
Thus, the final answer is \((\frac{1}{2}, 5)\).
Given equations:
\[
\begin{cases}
8x + 7y = 39 \quad \text{(1)} \\
4x - 14y = -68 \quad \text{(2)}
\end{cases}
\]
### Step 1: Multiply the First Equation
To eliminate the \(y\)-term, we can multiply the first equation (1) by \(2\) so that the coefficients of \(y\) in both equations will match in magnitude (since \(7\) in equation (1) becomes \(14\) after multiplication):
\[
2(8x + 7y) = 2(39) \\
16x + 14y = 78 \quad \text{(new equation 1)}
\]
### Step 2: Write the System of Equations
Now we have the modified system of equations:
\[
\begin{cases}
16x + 14y = 78 \quad \text{(new equation 1)} \\
4x - 14y = -68 \quad \text{(2)}
\end{cases}
\]
### Step 3: Add the Equations to Eliminate the \(y\)-terms
Now add the two equations together to eliminate the \(y\)-terms:
\[
(16x + 14y) + (4x - 14y) = 78 + (-68)
\]
This simplifies to:
\[
16x + 4x + 14y - 14y = 78 - 68
\]
\[
20x = 10
\]
### Step 4: Solve the New Equation for \(x\)
Now solve for \(x\):
\[
x = \frac{10}{20} = \frac{1}{2}
\]
### Step 5: Substitute the \(x\)-value into Either Original Equation
Now substitute \(x = \frac{1}{2}\) back into one of the original equations to find \(y\). Let's use equation (1):
\[
8\left(\frac{1}{2}\right) + 7y = 39
\]
This simplifies to:
\[
4 + 7y = 39
\]
Subtract \(4\) from both sides:
\[
7y = 39 - 4
\]
\[
7y = 35
\]
Now divide by \(7\):
\[
y = 5
\]
### Step 6: Check the Solution
To check, substitute \(x = \frac{1}{2}\) and \(y = 5\) into both original equations.
**For equation (1):**
\[
8\left(\frac{1}{2}\right) + 7(5) = 39 \\
4 + 35 = 39 \quad \text{(True)}
\]
**For equation (2):**
\[
4\left(\frac{1}{2}\right) - 14(5) = -68 \\
2 - 70 = -68 \quad \text{(True)}
\]
Both equations hold true with the given values.
### Final Answer
The solution to the system of equations is:
\[
\left(\frac{1}{2}, 5\right)
\]
Thus, the final answer is \((\frac{1}{2}, 5)\).
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