Problem 9 - Introductory Algebra for College Students

Question

Solve each equation. Be sure to check your proposed solution by substituting it for the variable in the original equation. $$7(2 x-1)=42$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 3.5\).

Step 1: Distribute the multiplier

First step is to distribute the multiplier on the left side of the equation. This means, multiply 7 with \(2x\) and also with \(-1\). After this, the equation becomes \(14x - 7 = 42\).

Step 2: Isolate the variable term

Next step is to isolate the term with the variable. This can be done by adding 7 to both sides of the equation. This gives \(14x = 49\).

Step 3: Solve for the variable

Lastly, divide both sides by 14 to find the value of \(x\). This gives \(x = 49/14 = 3.5\).

Step 4: Verify the solution

Substitute the found value for \(x\) into the original equation. That is, verify that \(7(2*(3.5)-1) = 42\). This simplifies to \(42 = 42\), which is true. Therefore, the solution is valid.

Key Concepts

Distributive Property

The Distributive Property is an essential concept when solving equations. It allows us to simplify expressions by distributing a single term across terms inside parentheses. In the equation \(7(2x - 1) = 42\), the distributive property is used to multiply both terms inside the parentheses by 7.

• Start by multiplying 7 with \(2x\), resulting in \(14x\).
• Then multiply 7 by \(-1\), which gives \(-7\).

After applying the distributive property, the equation simplifies to \(14x - 7 = 42\). This process breaks down a complex expression and helps to make the equation more manageable for further steps. The distributive property not only simplifies the solving process but is also crucial for working with algebraic expressions effectively.

Isolating Variable

Isolating the variable is a crucial step in solving equations, as it helps in finding the value of the unknown. In our equation, \(14x - 7 = 42\), the target is to have only \(x\) terms on one side.
To achieve this, we perform operations that eliminate other constants or variables:

• Add 7 to both sides to move the constant \(-7\) away from the \(x\) term. This changes the equation to \(14x = 49\).
• Next, divide both sides by 14 to isolate \(x\) completely. This results in \(x = 3.5\).

These steps ensure that \(x\) is isolated, making it possible to see its exact value clearly. Remember, the goal is always to perform inverse operations that maintain balance in the equation while getting \(x\) on its own.

Verification of Solutions

Verification is the final step that confirms the accuracy of the derived solution. After solving for \(x\), it's important to substitute the value back into the original equation to ensure it satisfies the entire equation.For the equation \(7(2x - 1) = 42\), substitute \(x = 3.5\) back:

• Calculate \(2 \times 3.5 = 7\).
• Then, \(7 - 1 = 6\).
• Finally, \(7 \times 6 = 42\).

Since both sides of the equation balance, \(42 = 42\), we verify that \(x = 3.5\) is indeed the correct solution. Verification gives confidence in your solution process and ensures no mathematical errors were made during the calculations.