Question

7
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$$\begin{cases} 2x = 8 - 4y \\\\ 17x - 31y = 3 \\end{cases}$$

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my calculations (grid)

8
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$$\begin{cases} 3y = -6x + 21 \\\\ 5x + 2y = 4 \\end{cases}$$

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my calculations (grid)

9 joe is 10 years older than three times his son’s age. the sum of their ages is 58. how old are joe and his son? my calculations (grid)

10 the length of a rectangle is 5 units more than twice its width. the perimeter of this rectangle is 220 units. what is the area of this rectangle? my calculations (grid)

Explanation:

Problem 7

Step 1: Simplify the first equation

From \( 2x = 8 - 4y \), divide by 2:
\( x = 4 - 2y \)

Step 2: Substitute \( x \) into the second equation

Substitute \( x = 4 - 2y \) into \( 17x - 31y = 3 \):
\( 17(4 - 2y) - 31y = 3 \)
Expand: \( 68 - 34y - 31y = 3 \)
Combine like terms: \( 68 - 65y = 3 \)

Step 3: Solve for \( y \)

Subtract 68: \( -65y = 3 - 68 = -65 \)
Divide by -65: \( y = 1 \)

Step 4: Solve for \( x \)

Substitute \( y = 1 \) into \( x = 4 - 2y \):
\( x = 4 - 2(1) = 2 \)

Problem 8

Step 1: Simplify the first equation

From \( 3y = -6x + 21 \), divide by 3:
\( y = -2x + 7 \)

Step 2: Substitute \( y \) into the second equation

Substitute \( y = -2x + 7 \) into \( 5x + 2y = 4 \):
\( 5x + 2(-2x + 7) = 4 \)
Expand: \( 5x - 4x + 14 = 4 \)
Combine like terms: \( x + 14 = 4 \)

Step 3: Solve for \( x \)

Subtract 14: \( x = 4 - 14 = -2 \)

Step 4: Solve for \( y \)

Substitute \( x = -2 \) into \( y = -2x + 7 \):
\( y = -2(-2) + 7 = 4 + 7 = 9 \)

Problem 9

Step 1: Define variables

Let son’s age = \( s \), Joe’s age = \( 3s + 10 \) (since Joe is 10 + 3×son’s age).

Step 2: Set up the equation

Sum of ages: \( s + (3s + 10) = 58 \)
Simplify: \( 4s + 10 = 58 \)

Step 3: Solve for \( s \)

Subtract 10: \( 4s = 48 \)
Divide by 4: \( s = 12 \)

Step 4: Find Joe’s age

Joe’s age: \( 3(12) + 10 = 36 + 10 = 46 \)

Problem 10

Answer:

\( x = 2 \), \( y = 1 \)