13) Three times Mary's age added to Beth's age is 34 years. Half of Mary's age minus Beth's age is 1 year. How old are Mary and Beth? 14) A rectangle is x inches wide and 3y inche... 13) Three times Mary's age added to Beth's age is 34 years. Half of Mary's age minus Beth's age is 1 year. How old are Mary and Beth? 14) A rectangle is x inches wide and 3y inches long. If the length plus twice the width is 12 inches and the length plus the width is 9 inches, what is the length? Read more

Steps to Solve

1. Define variables for Question 13

Let $M$ be Mary's age and $B$ be Beth's age.

2. Write the equations for Question 13

Based on the problem statement, we can write the following system of equations:

$$3M + B = 34$$ $$\frac{1}{2}M - B = 1$$

3. Solve for M in Question 13

We can solve this system by adding the two equations to eliminate $B$:

$$3M + B + \frac{1}{2}M - B = 34 + 1$$ $$3M + \frac{1}{2}M = 35$$ $$\frac{6}{2}M + \frac{1}{2}M = 35$$ $$\frac{7}{2}M = 35$$ $$M = 35 \times \frac{2}{7}$$ $$M = 10$$

4. Solve for B in Question 13

Substitute $M = 10$ into the first equation to solve for $B$:

$$3(10) + B = 34$$ $$30 + B = 34$$ $$B = 34 - 30$$ $$B = 4$$

5. Write the answer for Question 13

Mary is 10 years old and Beth is 4 years old.

6. Define variables for Question 14

Let $x$ be the width of the rectangle and $3y$ be the length of the rectangle.

7. Write the equations for Question 14

Based on the problem statement, we can write the following system of equations: $$3y + 2x = 12$$ $$3y + x = 9$$

8. Solve for x in Question 14 Subtract the second equation from the first equation to eliminate $3y$:

$$(3y + 2x) - (3y + x) = 12 - 9$$ $$3y + 2x - 3y - x = 3$$ $$x = 3$$

9. Solve for 3y in Question 14

Substitute $x = 3$ into the second equation to solve for $3y$:

$$3y + 3 = 9$$ $$3y = 9 - 3$$ $$3y = 6$$

10. Write the answer for Question 14

The length of the rectangle is 6 inches.