Problem 64 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve and check: \(6(y-1)+7=9 y-y+1\)

Step-by-Step Solution

Answer

The solution to the equation is y=0

Step 1: Simplify each side of the equation

Remove the parenthesis to result in this equation: 6y-6+7=9y-y+1

Step 2: Remove Like terms on right side of the equation

Combine like terms on right side of the equation: 6y-6+7=8y+1

Step 3: Subtract 6y from each side

Next, subtract 6y from each side of the equation to get: 1=2y+1

Step 4: Simplify by collecting like terms

We simplify our equation by subtracting 1 from both sides of the equation to get: 2y=0

Step 5: Solve for y

Finally, we divide each side by 2 to obtain y=0. Therefore, the solution to the equation is y=0.

Step 6: Check the solution

Substitute y with 0 in the original equation. If both sides of the equation are equal, then the solution is correct. After substitution, we get: 6(0-1)+7=9(0)-0+1 which simplifies to -6+7=1, which in turn simplifies to 1=1, which means our solution y=0 is correct.

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of letters, numbers, and arithmetic operations without an equality sign. These expressions represent quantities and allow us to perform algebraic manipulations. For example, in the exercise, 6(y-1)+7 and 9y - y + 1 are algebraic expressions representing each side of the equation.

Simplifying algebraic expressions is a crucial step in solving equations. It involves combining like terms and using the distributive property to eliminate parentheses. For instance, the expression 6(y-1) can be simplified by multiplying 6 by both y and -1, resulting in 6y - 6.

Like Terms in Algebra

Like terms in algebra are terms that have the same variables raised to the same power. Numbers can also be like terms if they have no variable attached. In the exercise, the simplification process includes finding like terms and combining them.

On the right side of the equation, 9y and -y are like terms because they both have the variable y without any exponent, so they can be combined to 8y. On the left side, the constants -6 and +7 are like terms and are combined to +1. Recognizing and combining like terms is essential for simplifying equations and moving towards a solution.

Checking Solutions in Algebra

Checking solutions is a fundamental step in algebra to ensure the accuracy of our answer. This process involves substituting the solution back into the original equation to verify that the equality holds true.

For the given exercise, the concluded solution is y=0. To check this solution, we substitute 0 for y into the original equation and simplify. If the resulting expressions on both sides of the equation are equal, our solution is correct. As demonstrated, substituting y=0 gives us an equality of 1=1, confirming that our solution is indeed correct. This verification step is crucial as it not only confirms the solution but also helps to prevent errors during the calculation process.