Problem 27 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve 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 is \(x = 7\).

Step 1: Simplify both sides

This step involves distributing the multiplication into the parentheses on both sides. This simplifies the expression to: -4 + 4x = 3x + 3.

Step 2: Combine like terms

To isolate x, subtract 3x from both sides, it becomes: -4 + x = 3.

Step 3: Solve for x

Finally, to further isolate x, add 4 to both sides which makes: x = 7.

Step 4: Verify the solution

To verify the solution substitute 7 for x in the original equation. If both sides of the equation remain equal then the solution is correct. In this case, the equation becomes \(6 = -4(1-7) + 3(7+1)\) which simplifies to \(6=6\), therefore x = 7 is the correct solution.

Key Concepts

Solving Linear Equations

Linear equations are mathematical expressions where each term is either a constant or the product of a constant and a single variable. The main goal in solving these equations is to find the value of the variable that makes the equation true. For the given exercise, the objective is to solve the equation \(6 = -4(1-x) + 3(x+1)\). To begin, you need to simplify both sides of the equation by distributing the numbers outside the parentheses through the terms inside the parentheses. This means multiplying as follows: \(-4(1-x)\) becomes \(-4 + 4x\) and \(3(x+1)\) becomes \(3x + 3\). After this step, the equation simplifies to \(-4 + 4x = 3x + 3\). This sets the stage for the next steps: combining like terms and isolating the variable.

Equation Verification

After finding a solution to an equation, it's crucial to verify it. This ensures that no mistakes were made during the solving process. Verifying an equation involves substituting the found value back into the original equation and checking if both sides are equal. In this exercise, after solving for \(x\) and finding \(x = 7\), we substitute 7 back into the original equation \(6 = -4(1-x) + 3(x+1)\). By replacing \(x\) with 7, the equation becomes \(6 = -4(1-7) + 3(7+1)\), which simplifies to \(6 = 6\). Since both sides of the equation are equal, the solution \(x = 7\) is confirmed to be correct. This step is essential as it double-checks your solution for accuracy.

Distribution Property

The distribution property is a crucial tool in solving equations. It allows you to eliminate parentheses by multiplying a single term across the terms inside the parentheses. For instance, in the expression \(-4(1-x)\), the distribution property helps simplify it to \(-4 + 4x\). This means you multiply \(-4\) with each term within the parentheses: \(-4\times1\) and \(-4\times-x\). Similarly, in \(3(x+1)\), it becomes \(3x + 3\) by multiplying 3 with each term in the parentheses. Using the distribution property ensures the equation is easier to handle by removing the parentheses, allowing you to proceed with combining like terms and simplifying the equation further.

Combining Like Terms

After distributing, you'll often have terms that need combining to simplify the equation. Combining like terms involves collecting all terms with the same variable on one side of the equation. For instance, in \(-4 + 4x = 3x + 3\), both \(4x\) and \(3x\) are terms involving the variable \(x\). By moving \(3x\) to the left side (done by subtracting \(3x\) from both sides), you simplify the terms to \(-4 + x = 3\), where both 'like terms' involving \(x\) are now on one side. Once like terms are combined, you can solve for the variable by isolating it, simplifying the problem to find the exact value of \(x\).