Problem 13 - College Algebra

Question

In Exercises \(1-16,\) solve and check each linear equation. $$ 16=3(x-1)-(x-7) $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 6\).

Step 1: Distribute

First, distribute the 3 and the negative sign across the expressions in parentheses. We get \(16 = 3x - 3 - x + 7\).

Step 2: Combine like terms

Combine like terms on the each side of the equation. On the right side, we combine \(3x\) and \(-x\) to get \(2x\), and we combine \(-3\) and \(7\) to make \(4\) . As result, the equation becomes \(16 = 2x + 4\)

Step 3: Isolate the variable

In order to solve for \(x\), subtract 4 from both sides of the equation and we find \(12 = 2x\)

Step 4: Solve for x

Dividing each side of the equation by 2, we find that \(x = 6\)

Step 5: Check the solution

Substitute \(x = 6\) back into the original equation. \(16 = 3(6 - 1) - (6 - 7)\). Simplifying this gives \(16 = 16\), hence our solution is correct.

Key Concepts

Solving Equations

Solving linear equations involves finding the value of the variable that makes the equation true. Linear equations typically take the form of \( ax + b = c \). Solving these means you'll go through a process to "undo" whatever is done to the variable. This often involves applying operations that reverse those in the equation, such as addition or subtraction, and multiplication or division. The goal is to simplify the equation step by step.

• Start with Distribution: Eliminate parentheses.
• Combine Like Terms: Simplify your equation by organizing similar items together.
• Isolate the Variable: Get your variable alone on one side of the equation.
• Check Your Solution: It's always wise to verify your solution is correct by plugging it back into the original equation.

Breaking down and solving equations builds foundational math skills essential for everything from more advanced algebra to real-world problem-solving.

Distributive Property

The distributive property is a fundamental tool in algebra when dealing with expressions inside parentheses. It allows you to simplify expressions by removing the parentheses and is expressed as \( a(b+c) = ab + ac \). This property helps to simplify the equation when you have a term outside the parentheses, as in our exercise.

• Apply the distributive rule by multiplying the coefficient by each term inside the parentheses.
• In our example, distribute \(3(x-1)\) to get \(3x - 3\), and \(-(x-7)\) to get \(-x + 7\).
• The expression becomes easier to handle and understand: \(16 = 3x - 3 - x + 7\).

Understanding and applying this property is a critical step in simplifying complex equations, transforming them into a form that can be further simplified by combining like terms.

Combining Like Terms

Combining like terms in an equation is about gathering all similar terms together to simplify an equation. This step reduces the equation's complexity and is crucial for the process of isolating the variable.

• Identify Similar Terms: Look for terms that have the same variable and power.
• Add or Subtract: Combine these terms by adding or subtracting coefficients. For instance, in the equation \(16 = 3x - 3 - x + 7\), you combine \(3x\) and \(-x\) to get \(2x\), and \(-3\) and \(7\) to get \(4\).
• Resulting Equation Becomes Simpler: After combining, the equation reads \(16 = 2x + 4\), easier to manage for isolation.

By consolidating, the equation becomes clearer and neater, setting the stage for isolating the variable.

Isolating Variables

Isolating the variable means rearranging the equation so that the variable is alone on one side of the equation, with a solution on the other side. Achieving this requires a series of logical and reverse operations.

• Eliminate Constants: Subtract or add numbers from both sides to move constants away from the variable. With \(16 = 2x + 4\), subtract \(4\) from both sides to get \(12 = 2x\).
• Divide or Multiply: If the variable is multiplied by a coefficient, divide both sides by that coefficient. Divide \(12\) by \(2\) to find \(x = 6\).
• Solution Checking: Substitute back to ensure the equation holds. Plug \(x = 6\) into the original equation and verify: \(16 = 16\). Success!

Mastering this step efficiently will aid in tackling more complicated equations and supports strong algebra skills that are applied in many areas of math and science.