Problem 51 - Algebra 1: Concepts and Skills

Question

Solve the equation. \(-x+(5 x-7)=-5\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = 0.5\)

Step 1: Combine like terms

First, simplify the left side of the equation by combining the terms involving 'x'. The left side equation becomes: \(4x - 7\)

Step 2: Isolate the term with the variable

The goal is to isolate 'x', so to do so, add 7 to both sides to balance the equation. That results in: \(4x = -5+7\)

Step 3: Solve for the variable

Simplify the right side to get: \(4x = 2\). Therefore, to solve for 'x', divide both sides by 4. The result is: \(x = 2 / 4\)

Key Concepts

Combining Like Terms

Understanding how to combine like terms is a fundamental skill in algebra that helps simplify equations and make them easier to solve. Like terms are terms that contain the same variable raised to the same power. For instance, in the equation \(-x + (5x - 7) = -5\), the terms -x and 5x are like terms because they both contain the variable x to the first power.

To combine them, simply add or subtract the coefficients (the numbers in front of the variables) and keep the variable unchanged. In our example, -1x (since x is the same as 1x) plus 5x equals 4x. So, the left side of the equation simplifies to 4x - 7. This step reduces the complexity of the equation, paving the way for the next steps in finding the solution.

Isolate the Variable

Once you have combined like terms and simplified the equation as much as possible, the next step is to isolate the variable. Isolating the variable means to get the variable on one side of the equation by itself. In our equation, we began with 4x - 7 on the left side. To isolate x, we want to reverse any operations that are being done to it. Since x is being multiplied by 4 and then reduced by 7, we first address the subtraction by adding 7 to both sides of the equation, leading to 4x = -5 + 7.

It is crucial to perform the same operation on both sides to maintain the balance of the equation, which is the core principle of algebra. This step takes us closer to finding the value of x because we now have an equation where x is only being multiplied by a number, without any additional terms to complicate things.

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. After combining like terms and isolating the variable, we often end up with a simple equation known as a linear equation. A linear equation is one where the variable is to the power of one, and it is graphed as a straight line. In the case of our problem, we have 4x = 2.

To solve for x, we now perform the last inverse operation, which is division in this case, because x is multiplied by 4. By dividing both sides by 4, we get x = 2 / 4. Simplifying this fraction gives us x = 1 / 2 or x = 0.5, which is the solution to the linear equation. The process of solving linear equations is foundational in algebra and is applicable to many different types of problems. Mastering this process allows students to approach more complex algebraic expressions with confidence.