Problem 27 - 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. $$6=-4(1-x)+3(x+1)$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 1\).

Step 1: Distribute the factors

First, distribute the -4 to (1-x) and the 3 to (x+1). This leads to the following simplified equation: \[-4 + 4x + 3x + 3 = 6.\]

Step 2: Combine like terms

Combine the 'x' terms and the constant terms on the left side of the equation to further simplify. This provides the following equation: \[7x + -1 = 6.\]

Step 3: Isolate 'x'

To isolate 'x', add 1 to both sides, yielding: \[7x = 7.\] Then, divide by 7 to solve for 'x': \[x = 1.\]

Step 4: Verification

Substitute x = 1 into the original equation: \[6 =-4(1-1)+3(1+1),\] simplifying which gives 6 = 6, confirming the solution is correct.

Key Concepts

Solving Equations

Solving equations means finding the value of the variable that makes the equation true. In simpler terms, it involves determining which number, when substituted for the variable, will satisfy the equation.
This process turns algebraic expressions into simpler forms to find the solution.
The key steps in solving any equation include:

• Simplifying both sides, if necessary.
• Identifying variables and constants.
• Isolating the variable to one side of the equation.

By systematically following these steps, you can unravel even the most complex algebraic problems.

Distribution Property

The distribution property is a fundamental algebraic rule that helps to simplify equations during the solving process.
It involves multiplying a single term by each term inside a set of parentheses.In the context of our exercise, this means taking the expression \(-4(1-x)+3(x+1)\) and applying distribution:

• Multiply \(-4\) by \((1-x)\), yielding \(-4 + 4x\).
• Multiply \(3\) by \((x+1)\), resulting in \(3x + 3\).

This property helps expand expressions, clearing the way for easier manipulation of terms.

Combining Like Terms

Combining like terms is all about simplifying expressions.
It involves adding or subtracting coefficients of terms that have the same variable or are constant.After using the distribution property in our example, we have:

• Combine \(4x + 3x\) to get \(7x\).
• For constants, combine \(-4 + 3\) to result in \(-1\).

By reducing the number of terms, combining like terms simplifies the equation greatly, moving us closer to finding the variable's value.

Isolation of Variables

Isolation of variables is the technique of manipulating an equation to get the variable alone on one side.
This step is crucial as it directly leads to finding the solution to the equation.In the equation \(7x + -1 = 6\),we first add \(1\) to both sides: \(7x = 7\). Then, we divide both sides by \(7\) to get \(x = 1\).
This complete separation of the variable allows us to easily identify the specific value that satisfies the equation. Verification, a quick substitution back into the original equation, ensures the solution is correct, reinforcing understanding and accuracy.