Problem 54 - Introductory Algebra for College Students

Question

Solve each equation using the addition property of equality. Be sure to check your proposed solutions. $$7 x+3=6(x-1)+9$$

Step-by-Step Solution

Answer

The solution to the equation \(7x+3=6(x-1)+9\) is \(x = 0\).

Step 1: Simplify the equation

The best approach is to simplify both sides of the equation as much as possible before starting to solve for \(x\). Simplify the right-hand side of the equation first by multiplying \(6\) times \(x-1\). Hence, the equation becomes: \(7x+3 = 6x-6 +9\).

Step 2: Combine like terms

After simplifying, combine the like terms on both sides of the equation. This results in: \(7x+3 = 6x+3 \)

Step 3: Solve for \(x\)

Subtract \(6x\) from both sides of the equation to isolate the variable on one side. After doing so, the equation becomes: \(x =0\). This is the final solution.

Step 4: Check the proposed solution

Substitute the found solution, \(x = 0\), back into the original equation to verify that it holds true. The original equation is \(7(0)+3=6(0-1)+9\), which simplifies to \(3 =3\). This checks out, indicating that the found solution \(x = 0\) is in fact correct.

Key Concepts

Simplifying Equations

Simplifying equations is all about making a complex expression easier to solve. In this context, let's consider the equation from the exercise:

• The left side is already simplified: \(7x + 3\)
• Focus on simplifying the right side: \(6(x - 1) + 9\)

To simplify, distribute the 6 across the terms in the parenthesis:

• Multiply \(6\) by \(x\): \(6x\)
• Multiply \(6\) by \(-1\): \(-6\)
• Add \(9\) back: so, \(6x - 6 + 9\)

Now, the equation is easier to handle: \(7x + 3 = 6x + 3\). Simplification forms the foundation for finding solutions efficiently. Breaking down complex expressions into simpler ones is a crucial first step.

Combining Like Terms

Once the equation is simplified, the next step is to combine like terms. This means gathering similar variable terms and constants together. In our example:

• The left hand side has \(7x\) and \(+3\).
• The right hand side has \(6x\) and \(+3\).

Combining like terms ensures each variable is grouped neatly to further simplify the equation.

• On the left side: No like terms to combine, \(7x + 3\)
• On the right side: \(6x + 3\)

Both sides already reflect like terms. Recognizing and arranging like terms is vital since it enables straightforward manipulation of the equations.

Solving Linear Equations

After simplifying and combining like terms, solving linear equations becomes a straightforward task. In the problem at hand, our equation is:

• \(7x + 3 = 6x + 3\)

Our goal is to isolate the variable \(x\). Begin by eliminating \(6x\) from both sides:

• Subtract \(6x\) from \(7x\): \(7x - 6x = x\)
• The equation simplifies to: \(x + 3 = 3\)

Next, remove 3 from both sides to solve for \(x\):

• Subtract 3: \(x = 0\)

This is the value of \(x\) that solves the equation. Solving linear equations is a systematic approach where we aim to get the variable by itself, confirming the solution is valid by plugging \(x = 0\) back into the original equation, proving our solution indeed satisfies the equation.