Problem 22 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve equation. Be sure to check your proposed solution by substituting it for the variable in the original equation. \(8(y+3)=3(2 y+12)\)

Step-by-Step Solution

Answer

The solution to the equation is \(y = 6\)

Step 1: Distribute Multiplication

Distribute the multiplication within parentheses on both sides of the equation to simplify: \(8y + 24 = 6y + 36\)

Step 2: Combine Like Terms

Move the y terms to left side and constants to the other side: \(8y - 6y = 36 - 24\) which simplifies to \(2y = 12\)

Step 3: Isolate the Variable

To isolate y, divide both sides of the equation by 2: \(y = 12 / 2\)

Step 4: Find the Solution

Perform the division to find the value of y: \(y = 6\)

Step 5: Check the Solution

Substitute the solution y = 6 back into the original equation to verify: \(8(6+3) = 3(2*6+12) => 72 = 72\) is true, so y = 6 is the correct solution

Key Concepts

Equation Simplification

Equation simplification is an important first step in solving linear equations. When you simplify an equation, you aim to make it easier to work with. That means reducing it to a form where it’s easier to see what’s happening and to solve for the unknown variable. Simplifying often involves combining like terms and eliminating unnecessary parts of the equation.

In the original equation given by the problem, we see terms like \(8(y+3)\) and \(3(2y+12)\). Before any further steps can be taken, these expressions need to be simplified. This is where processes like combining together terms that are the same, such as adding up all the terms that contain \(y\), come into play. Also, simplifying means getting rid of extra numbers or terms that do not have a variable attached to them by performing basic arithmetic.

Distributive Property

The distributive property is a key algebraic rule used in the problem above, particularly in the first step. This property allows you to remove parentheses by distributing a multiplication across the terms within the parentheses. Here's how it works: If you have an expression like \(a(b+c)\), you distribute the \(a\) to both \(b\) and \(c\), ending up with \(ab+ac\).

Applying the distributive property to the original equation gives \(8(y+3) = 8y + 24\) and \(3(2y+12) = 6y + 36\). Each term inside the parentheses got multiplied separately by the number outside. By distributing, you turn complex expressions into simpler equations. This makes it easier to move on with the next steps, like combining terms or isolating variables.

Variable Isolation

Variable isolation is the ultimate goal when solving an equation. It involves getting the variable you are solving for on one side of the equation by itself. This means you can directly calculate its value. Once simplified, equations often look like \(ax = b\), where \(a\) is a number and \(x\) is the variable you need to find.

In the step-by-step solution, we see this in action. After distributing and combining like terms, you have an equation like \(2y = 12\). Isolating the variable \(y\) involves dividing both sides of the equation by 2 to solve for \(y\). This gives \(y = 6\). Always remember to perform the same operation on both sides of the equation in order to keep it balanced. This is how you accurately find the value of your variable. Checking your solution by substituting back in the original equation is also a part of successfully isolating and verifying the variable solution.