Problem 51 - Introductory Algebra for College Students

Question

Solve each equation using the addition property of equality. Be sure to check your proposed solutions. $$12-6 x=18-7 x$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 6\).

Step 1: Isolate the variable

The first step in this exercise is to isolate the variable. To do this, add \(7x\) to both sides of the equation to eliminate \(-7x\) on the right hand side. The equation then becomes: \(12-6x+7x=18\). We simplify to get \(x = 18 - 12\).

Step 2: Solve for x

Next, you simplify the equation to solve for \(x\). When we subtract \(12\) from both sides, we get \(x = 18 - 12 = 6\). So, \(x = 6\).

Step 3: Check the solution

Lastly, after finding the solution, you then substitute \(x = 6\) into the original equation to verify if it’s correct. Substituting \(x = 6\) into the original equation \(12 - 6x = 18 - 7x\) gives \(12 - 36 = 18 - 42\). Simplifying both sides gives \(-24 = -24\), so the solution is correct.

Key Concepts

Solving Linear Equations

Understanding how to solve linear equations is a foundational skill in algebra. A linear equation is an equation where the highest power of the variable is one. To solve such equations, the goal is to find the value of the variable that makes the equation true. This often involves a series of steps that simplify the equation until the variable is by itself on one side of the equality.

The process often begins by applying the addition property of equality, which states that you can add the same value to both sides of an equation without changing the equation's solution. This can help to combine like terms or move terms from one side of the equation to the other. It's important to perform the same operations to both sides to maintain the balance of the equation - just as you would to keep a scale in balance.

Another useful step is to simplify both sides of the equation as much as possible. This could involve combining like terms, or simplifying expressions by carrying out addition, subtraction, multiplication, or division. By simplifying both sides, it can make the path to the solution clearer and easier to reach.

Isolate the Variable

To find the solution to a linear equation, you need to isolate the variable, meaning you want it alone on one side of the equal sign. This makes it simple to see what the variable equals. The process of isolating the variable may involve several operations, including adding, subtracting, multiplying, or dividing both sides of the equation by the same number.

Use of Inverse Operations

When you isolate the variable, you typically use inverse operations. For example, if a variable is being subtracted in an equation, you would add the same number to both sides to negate this subtraction. Similarly, if the variable is being divided by a number, you would multiply both sides by that number to eliminate the division.

A key concept here is the need for balance: whatever operation you do to one side, you must do to the other. This ensures that the two sides of the equation remain equal, which is pivotal when you are working towards the correct solution for the variable in question.

Checking Algebraic Solutions

Once a proposed solution for an equation is found, it's essential to verify its correctness. This is a step that should not be skipped, as simple mistakes can lead to incorrect solutions. To check a solution, you substitute the value back into the original equation and see if the equation holds true.

For example, if you think the variable 'x' equals 6, you would replace 'x' with 6 throughout the equation and simplify both sides to make sure the equality is true. If both sides of the equation have the same value after substitution, the solution is correct.

Why Checking Matters

Checking not only ensures that your solution is correct but also helps to confirm that there were no arithmetic errors along the way. This step reinforces your understanding of the equation and can also provide insight into the relationship between the variables and numbers within the equation, deepening your mathematical comprehension and fluency.