Problem 4 - 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. \(3 x+2 x+64=40-7 x\)

Step-by-Step Solution

Answer

The solution to the equation \(3x + 2x + 64 = 40 - 7x\) is \(x = -2\)

Step 1: Combine Like Terms

On the left-hand side of the equation, combine the \(3x\) and \(2x\) terms together. That gives us \(5x + 64 = 40 - 7x\)

Step 2: Rearrange the Equation

Move the \(x\) terms to one side of the equation and the constant terms to the other side. We get \(5x + 7x = 40 - 64\)

Step 3: Simplify the Equation

Simplify both sides of the equation. We get \(12x = -24\). Now, solve for \(x\) by dividing each side by 12. This gives us \(x = -2\)

Step 4: Check the Solution

Substitute the value of \(x\) (which is -2) back into the original equation. We get \(3(-2) + 2(-2) + 64 = 40 - 7(-2)\). Simplifying each side, we have \(58 = 58\), which validates our solution

Key Concepts

Algebraic Expressions

When working with algebra, you'll often encounter algebraic expressions. These are mathematical phrases that include numbers, variables (like x), and operation symbols such as plus, minus, multiplication, and division.

Creating and understanding algebraic expressions is a fundamental skill in solving linear equations. For instance, in the exercise (3x + 2x + 64 = 40 - 7x), 3x, 2x, and -7x are algebraic expressions that represent the quantity of 'x' multiplied by 3, 2, and -7, respectively. The numbers without variables, here 64 and 40, are constants and are also part of the algebraic expression.

To solve for 'x', you need to manipulate these expressions, combining like terms and rearranging them to isolate the variable. By mastering the language of algebraic expressions, you'll be well equipped to tackle a wide range of algebra problems.

Equation Simplification

Simplification is key to solving linear equations effectively. Equation simplification involves reducing the equation to its simplest form to make it easier to solve. This often includes combining like terms and eliminating unnecessary parts of the equation.

In our example, combining like terms (such as 3x and 2x on one side and moving all 'x' terms to one side) is the first step in simplification. The equation went from 3x + 2x + 64 = 40 - 7x to 5x + 64 = 40 - 7x and then to 12x = -24 after rearranging.

Simplifying an equation makes it clearer, reduces potential mistakes, and often reveals the solution more directly. It's a crucial process that, when practiced, becomes second nature to those studying algebra.

Substitution Method

The substitution method is a powerful tool for verifying the solutions to equations. Once you believe you have solved the equation, substituting the value back into the original equation can either confirm or disprove your solution.

In our practice problem, after simplifying, we found that x = -2. To check this, we substitute -2 back into the original equation for every instance of 'x'. This gives us 3(-2) + 2(-2) + 64 on the left and 40 - 7(-2) on the right. Simplifying both sides should yield the same value if the solution is correct.

After substitution and simplification, both sides equaled 58, confirming that x = -2 is indeed the correct solution. The substitution method doesn't just provide validation; it also reinforces understanding of algebraic manipulation and the relationship between equations and their solutions.