Problem 26 - 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. \(7(3 x-2)+5=6(2 x-1)+24\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = 27 / 9 = 3\).

Step 1: Apply the distributive property of multiplication

Let's distribute the multiplication across the terms in parentheses: \(21x - 14 + 5 = 12x - 6 + 24\). Then simplify it: \(21x - 9 = 12x + 18\)

Step 2: Isolate the variable

Let's move all terms involving x to the left side and constants to the right side to isolate x: \(21x - 9 - 12x = 12x + 18 - 12x\), which simplifies to: \(9x = 27\)

Step 3: Solve for x

Finally, we'll divide both sides by 9 so that x remains on its own on the left side, resulting in: \(x = 27 / 9\)

Step 4: Check the solution

Substitute x = 3 into the original equation to verify the solution.

Key Concepts

Distributive Property

The distributive property is a foundational principle in algebra that allows us to simplify equations by distributing multiplication over addition or subtraction inside parentheses. In simpler terms, when you have something like \(a(b + c)\), you multiply both \(b\) and \(c\) by \(a\), resulting in \(ab + ac\).
In the original exercise, we apply the distributive property to the terms \(7(3x-2)\) and \(6(2x-1)\).
1. For \(7(3x - 2)\), we distribute the 7: \(7 \times 3x = 21x\) and \(7 \times (-2) = -14\). So, it becomes \(21x - 14\).
2. For \(6(2x - 1)\), similarly, we have \(6 \times 2x = 12x\) and \(6 \times (-1) = -6\). So, it becomes \(12x - 6\).
This property helps in setting up the equation in a form that's easier to work with, paving the way for isolating variables.

Isolating Variables

Isolating the variable involves rearranging the equation so that the variable is alone on one side of the equation. This process is quite straightforward once you use basic arithmetic operations.
In our example, the equations after using the distributive property look like \(21x - 9 = 12x + 18\).
We can isolate \(x\) by performing the following steps:

• Subtract \(12x\) from both sides to move all terms involving \(x\) to the left, leading to \(21x - 12x - 9 = 18\).
• This simplifies to \(9x - 9 = 18\).
• Add 9 to both sides to isolate terms involving \(x\), resulting in \(9x = 27\).
• Finally, divide both sides by 9 to solve for \(x\). This results in \(x = 3\).

Isolating variables is key to solving any equation as it allows us to pinpoint the value of the variable in question.

Checking Solutions

Checking solutions is a vital step in solving equations, as it verifies the accuracy of our result. It ensures no errors were made during calculations or algebraic manipulations.
To check a solution, substitute the value of the variable back into the original equation and see if both sides of the equation equal.
In our exercise, plug \(x = 3\) back into the original equation: \(7(3x-2)+5=6(2x-1)+24\).
1. Replace \(x\) with 3: \(7(3 \times 3 - 2) + 5 = 6(2 \times 3 - 1) + 24\).
2. Simplify both sides:

• Left side: \(7(9 - 2) + 5 = 7 \times 7 + 5 = 49 + 5 = 54\).
• Right side: \(6(6 - 1) + 24 = 6 \times 5 + 24 = 30 + 24 = 54\).

Both sides equal 54, confirming that our solution \(x = 3\) is indeed correct. This step not only confirms correctness but also boosts confidence in handling algebraic equations.