Problem 47 - Introductory Algebra for College Students

Question

Solve each equation using the addition property of equality. Be sure to check your proposed solutions. $$7-5 x+8+2 x+4 x-3=2+3 \cdot 5$$

Step-by-Step Solution

Answer

The solution to the equation is x = -5.

Step 1: Rewrite and Simplify the Left Hand Side

Combine like terms on the left hand side of the equation: \[7 - 5x + 8 + 2x + 4x - 3 = -x + 12\]

Step 2: Simplify the Right Hand Side

Solve for the right hand side: \[2 + 3 \cdot 5 = 17\]

Step 3: Set Left Hand Side Equal to Right Hand Side

Set the simplified left side equal to the right side: \[-x + 12 = 17\]

Step 4: Solve for 'x'

Add 'x' to both sides to isolate the variable on one side and simplify to solve for 'x': \[x = -5\]

Step 5: Check the Solution

Plug 'x = -5' back into the original equation to verify that the solution is correct: \[7 - 5(-5) + 8 + 2(-5) + 4(-5) - 3 = 2 + 3 \cdot 5\] Both sides of the equation equal 17. Therefore, 'x = -5' is the correct solution.

Key Concepts

Equation Solving

Equation solving is a fundamental concept in algebra where we find the value of the unknown variable that satisfies the equation. In this exercise, we are tasked with solving an equation using the addition property of equality. This property asserts that if we add the same quantity to both sides of an equality, the equality still holds. The goal is to gradually isolate the variable, here represented as \(x\). In the given equation, we start by simplifying each side step-by-step.

• First, simplify or combine like terms on one side of the equation.
• Then, solve the arithmetic on the other side.

By performing these calculations, we can isolate \(x\) to find its value. This systematic approach ensures accuracy and helps order the operations neatly. Observing order of operations and maintaining equality is essential for accurate results.

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and operators. They represent mathematical statements or real-world phenomena. Understanding how to work with algebraic expressions is key when solving equations. In our exercise example, the original equation contains terms such as \(-5x\), \(2x\), and \(4x\), making it crucial to recognize and combine like terms effectively. Like terms have the same variable part raised to the same power.

• Combine \(-5x\), \(2x\), and \(4x\) to get \(x\).
• For constant terms, add or subtract them together as needed.

Handling these expressions properly makes solving the rest of the equation more manageable and leads us to the true value of \(x\). The manipulation of these expressions is at the heart of algebra and key to unraveling complex equations.

Checking Solutions

After solving an equation, it's always a good practice to check if the solution is correct. This process involves substituting the found value back into the original equation. For instance, once we determined \(x = -5\), we substitute it back to verify our steps:

• Replace \(x\) with \(-5\) on both sides of the original equation.
• Calculate the results to see if both sides are equal.

In our specific example, substituting \(x = -5\) yielded the same result on both sides of the equation, confirming the accuracy of our solution. Checking solutions is crucial as it helps discover errors that may have occurred during calculations. It ensures confidence in our results and reinforces our understanding of how each component in the equation interacts.