Problem 45 - Introductory Algebra for College Students

Question

Solve each equation using the addition property of equality. Be sure to check your proposed solutions. $$6 y+3-5 y=14$$

Step-by-Step Solution

Answer

The solution to the equation \(6y + 3 - 5y = 14\) is \(y = 11\).

Step 1: Simplify the Equation

First, combine the 'y' terms on the left side of the equation. This gives \(6y - 5y + 3 = 14\), which simplifies to \(y + 3 = 14\).

Step 2: Isolate 'y'

Next, isolate 'y' by subtracting '3' from both sides of the equation. This gives \(y = 14 - 3\), which simplifies to \(y = 11\).

Step 3: Check the Solution

Finally, to check your solution, substitute \(y = 11\) back into the original equation. This yields \(6(11) + 3 - 5(11) = 14\), which simplifies to \(14 = 14\). Since both sides of the equation are equal, the solution \(y = 11\) is correct.

Key Concepts

Solving Linear Equations

Solving linear equations is a fundamental skill in algebra. At its core, it involves finding the value of the unknown variable that makes the equation true. A linear equation typically comes in the form ax + b = c, where x represents the unknown variable, and a, b, and c are constants.

To solve a linear equation, you often utilize the addition property of equality. This property states that if you add or subtract the same number from both sides of an equation, it remains balanced. For example, if you have an equation like 6y + 3 - 5y = 14, the first step is to simplify it by combining like terms; here, combine 6y and -5y to get y.

Once simplified, isolate the variable by performing inverse operations, such as addition, subtraction, multiplication, or division. In our case, subtract 3 from both sides to find the value of y. This process helps maintain the equality while inching closer to the solution.

Simplifying Equations

Simplifying equations is a crucial step when solving them, as it reduces complexity. The goal in simplification is to make the equation more straightforward by combining like terms and performing basic arithmetic operations. This involves a few key steps:

• **Combine Like Terms:** Gather all terms containing the same variable on one side. In our problem, combine 6y and -5y to simplify the expression to y.
• **Perform Operations:** Carry out any addition or subtraction to further simplify. For instance, in the equation y + 3 = 14, subtract 3 from both sides.
• **Reduce the Equation:** After performing these operations, you'll have a simplified equation like y = 11, which is easier to interpret and solve.

By focusing on simplifying each part of the equation step-by-step, you make solving it straightforward and less prone to mistakes. Employing the addition property of equality throughout this process can help ensure you maintain a correct and balanced equation.

Checking Solutions

Checking your solution is an essential step in solving equations. This confirms that the solution you found actually satisfies the original equation. Here’s how you can check your work effectively:

After finding the value for the variable, such as y = 11, substitute it back into the original equation to verify its correctness. In mathematical terms, replace y with 11 in 6y + 3 - 5y = 14.

Calculate: 6(11) + 3 - 5(11). Break it down further:

• Multiply: 6 times 11 equals 66.
• Do Subtraction: 5 times 11 equals 55.
• A simple arithmetic: Add and subtract, 66 + 3 - 55 results in 14.

Since both sides equal 14, the solution y = 11 is indeed correct. This checking step not only solidifies your understanding but also ensures accuracy in your final answer, giving you confidence in your solution.