Solve 7 + y = 5(2y – 1) + 3y. Check your solution.

Answer 1

To solve the equation \( 7 + y = 5(2y - 1) + 3y \), let's start by simplifying the right side.

1. Distribute the \( 5 \) on the right-hand side: \[ 5(2y - 1) = 10y - 5 \] So, the equation now looks like: \[ 7 + y = 10y - 5 + 3y \]

2. Combine the terms on the right-hand side: \[ 10y + 3y = 13y \] Thus, the equation simplifies to: \[ 7 + y = 13y - 5 \]

3. Next, we will isolate \( y \). First, subtract \( y \) from both sides: \[ 7 = 13y - y - 5 \] This simplifies to: \[ 7 = 12y - 5 \]

4. Add \( 5 \) to both sides: \[ 7 + 5 = 12y \] This gives: \[ 12 = 12y \]

5. Now divide both sides by \( 12 \): \[ y = 1 \]

Now that we have \( y = 1 \), let’s check the solution by substituting \( y \) back into the original equation.

The original equation is: \[ 7 + y = 5(2y - 1) + 3y \]

Substituting \( y = 1 \):

• Left side: \[ 7 + 1 = 8 \]

• Right side: \[ 5(2(1) - 1) + 3(1) = 5(2 - 1) + 3 = 5(1) + 3 = 5 + 3 = 8 \]

Both sides are equal, as \( 8 = 8 \).

Thus, the solution is confirmed: \[ \boxed{1} \]

Answer 2

110=2l+(2l-5)

Answer 3

To solve the equation \( 110 = 2l + (2l - 5) \), let’s simplify and solve for \( l \).

1. Start by distributing on the right side: \[ 110 = 2l + 2l - 5 \]

2. Combine like terms: \[ 110 = 4l - 5 \]

3. Next, add \( 5 \) to both sides to isolate the term with \( l \): \[ 110 + 5 = 4l \] Which simplifies to: \[ 115 = 4l \]

4. Now, divide both sides by \( 4 \): \[ l = \frac{115}{4} = 28.75 \]

Thus, the solution is \[ \boxed{28.75}. \]

Answer 4

Callie’s dad is installing a fence in the rectangular backyard. He purchased 110 feet of fencing and plans to use all the fencing. The width of the fenced area will be 5 feet less than twice the length of the fenced area. What is the length of the fenced area?

Answer 5

Let the length of the fenced area be denoted as \( l \) (in feet).

According to the problem, the width \( w \) can be expressed in terms of the length as: \[ w = 2l - 5 \]

The perimeter \( P \) of a rectangle can be given by the formula: \[ P = 2l + 2w \]

Since Callie's dad has purchased 110 feet of fencing, we have: \[ 2l + 2w = 110 \]

Now we can substitute \( w \) in terms of \( l \) into the perimeter equation:

1. Substitute \( w = 2l - 5 \) into the perimeter equation: \[ 2l + 2(2l - 5) = 110 \]

2. Simplify the equation: \[ 2l + 4l - 10 = 110 \] \[ 6l - 10 = 110 \]

3. Add \( 10 \) to both sides: \[ 6l = 120 \]

4. Divide both sides by \( 6 \): \[ l = 20 \]

Now that we have determined the length \( l \), let's find the width \( w \) to ensure it fits the original conditions:

• Calculate \( w \): \[ w = 2l - 5 = 2(20) - 5 = 40 - 5 = 35 \]

Now, double-check the perimeter using \( l \) and \( w \):

• Perimeter: \[ P = 2l + 2w = 2(20) + 2(35) = 40 + 70 = 110 \]

Thus, the calculations confirm the perimeter is indeed correct.

The length of the fenced area is \[ \boxed{20} \text{ feet.} \]