Question

problem #5: if 4 times a number is increased by 18, the result is 10 less than 6 times the number. what is the number?
what are you trying to find out: what is the number
workspace:
solution:
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problem #6: below the perimeter of the rectangle is equal to the perimeter of the triangle. using an equation, find the perimeter of both figures.
what are you trying to find out:
workspace:
solution:

Explanation:

Problem #5

Step1: Define the variable

Let the number be \( x \).

Step2: Translate the problem into an equation

4 times the number is \( 4x \), increased by 18 is \( 4x + 18 \). 10 less than 6 times the number is \( 6x - 10 \). So the equation is \( 4x + 18 = 6x - 10 \).

Step3: Solve the equation

Subtract \( 4x \) from both sides: \( 18 = 2x - 10 \).
Add 10 to both sides: \( 28 = 2x \).
Divide both sides by 2: \( x = 14 \).

Step1: Recall the perimeter formulas

Perimeter of a rectangle: \( P_{rectangle} = 2\times (length + width) \).
Perimeter of a triangle: \( P_{triangle} = sum\ of\ all\ sides \).

Step2: Calculate the perimeter of the rectangle

Length of the rectangle is \( 2x + 3 \), width is \( x - 1 \). So \( P_{rectangle} = 2\times((2x + 3)+(x - 1)) = 2\times(3x + 2) = 6x + 4 \).

Step3: Calculate the perimeter of the triangle

Sides of the triangle are \( x - 5 \), \( 3x - 2 \), and \( 4x - 3 \). So \( P_{triangle}=(x - 5)+(3x - 2)+(4x - 3)=8x - 10 \).

Step4: Set the perimeters equal and solve for \( x \)

Since \( P_{rectangle}=P_{triangle} \), we have \( 6x + 4 = 8x - 10 \).
Subtract \( 6x \) from both sides: \( 4 = 2x - 10 \).
Add 10 to both sides: \( 14 = 2x \).
Divide by 2: \( x = 7 \).

Step5: Find the perimeter

Substitute \( x = 7 \) into \( P_{rectangle}=6x + 4 \): \( 6\times7 + 4 = 42 + 4 = 46 \). (We can also check with the triangle's perimeter: \( 8\times7 - 10 = 56 - 10 = 46 \))

Answer:

The number is 14.

Problem #6