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.

Question

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?

Answer 1

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).

Answer 2

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

Answer 3

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.

Answer 4

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?

Answer 5

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: \[ x = 4, \quad y = 6 \]

Answer 6

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)

Answer 7

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!

Answer 8

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?

Answer 9

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.

Answer 10

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 (__
,
__).

Answer 11

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 \((\frac{1}{2}, 5)\).