Problem 73 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. (Some of the equations have no solution.) $$7(y+7)=5 y+59$$

Step-by-Step Solution

Answer

The solution to the equation is \(y= 5\).

Step 1: Expand the Bracket

To start the process, first distribute the '7' across the bracket on the left-hand side of the equation. This will give us: \(7y + 49 = 5y + 59\).

Step 2: Combine Like Terms

To simplify the equation, the same types of terms should be brought together. Subtract '5y' from both sides, to send it over to the left-hand side of the equation. Which gives \(2y + 49 = 59\). Next, subtract '49' from both sides to move it to the right-hand side. This results in: \(2y = 10\)

Step 3: Solve for y

Divide both sides of the equation by '2' to isolate 'y'. This gives: \(y = 5\) as the solution to the equation.

Step 4: Check the solution

Substitute '5' back into the original equation and simplify, this should give a true statement. \(7(5+7) =5 * 5 + 59\) simplifies to \(84 = 84\), which is a valid equation.

Key Concepts

Expand Brackets

When solving equations like the given problem, the first step often involves expanding the brackets. This means multiplying each term inside the bracket by the number or term outside. In this exercise, we have the equation:

• \(7(y + 7) = 5y + 59\)

To expand the brackets on the left side, we distribute the '7' across both terms inside the bracket, yielding:

• \(7 \cdot y + 7 \cdot 7 = 5y + 59\)
• Which simplifies to \(7y + 49 = 5y + 59\).

Expanding brackets is essential as it converts a more complex expression into a simpler form that is easier to manage.

Combine Like Terms

Once the brackets are expanded, we need to simplify the equation by combining like terms. Like terms are expressions that contain the same variables raised to the same power. In our example, after expanding, we have:

• \(7y + 49 = 5y + 59\)

To combine like terms, move the terms with 'y' on one side and the constants on the other.

• Subtract '5y' from both sides to get: \(2y + 49 = 59\).
• Then, subtract '49' from both sides, resulting in: \(2y = 10\).

Combining like terms allows you to simplify the equation further, making it easier to solve for the variable.

Isolate Variable

Now we focus on isolating the variable 'y'. Isolating the variable means rearranging the equation so that 'y' is on one side, with everything else on the other side. At this stage, our equation is:

• \(2y = 10\)

To isolate 'y', divide both sides of the equation by 2:

• \(y = \frac{10}{2}\)
• This simplifies to \(y = 5\).

Isolating the variable is crucial as it provides the solution to the equation. It's the step where we actually "solve" for the unknown.

Check Solution

After finding a solution, it's important to verify it by checking if it satisfies the original equation. Substitute the value of 'y' back into the original equation:

• \(7(y + 7) = 5y + 59\)
• Substitute '5' for 'y': \(7(5 + 7) = 5 \times 5 + 59\)

Calculate both sides:

• Left side: \(7 \times 12 = 84\)
• Right side: \(25 + 59 = 84\)

Since both sides are equal, \(84 = 84\), the solution is verified as correct. Checking your solution helps ensure the accuracy of your work and gives confidence in your solution.