Problem 45 - Elementary Algebra

Question

Solve. $$ -5 x+9=8 $$

Step-by-Step Solution

Answer

x = \(\frac{1}{5}\)

Step 1: Isolate the Variable Term

Subtract 9 from both sides of the equation to remove the constant term on the left side. This gets us closer to solving for the variable:\[-5x + 9 - 9 = 8 - 9\]Simplifying both sides gives us:\[-5x = -1\]

Step 2: Solve for the Variable

Now that the equation is \(-5x = -1\), divide both sides by \(-5\) to solve for \(x\):\[x = \frac{-1}{-5}\]Simplifying, we get:\[x = \frac{1}{5}\]

Key Concepts

Solving Linear Equations

Solving linear equations is one of the foundational skills in algebra. When faced with a linear equation, you're working to find the value of the variable that makes the equation true. A linear equation is any equation that can be written in the form: \( ax + b = c \). Here, \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable we're solving for.

To solve a linear equation, you may need to perform several operations:

• Addition or subtraction to move constant terms to the other side of the equation.
• Multiplication or division to isolate the variable and solve for it.

It's important to perform the same operation on both sides to maintain the balance of the equation. Otherwise, you risk altering the solution.

One-Variable Equations

One-variable equations are equations that contain only one unknown value or variable. In our original problem, the equation \(-5x + 9 = 8\) is a good example. We only need to find the value of \( x \) that makes the equation true.

By working through the given steps, we focus exclusively on manipulating expressions involving just this single variable. This process makes tackling the equation simpler. Once isolated, solving for the variable often involves a straightforward calculation, such as division or multiplication.

• Identify the variable and aim to isolate it on one side.
• Ensure that all constant terms are on the opposite side of the equation from the variable.
• Double-check your operations to ensure the equation remains balanced.

Successfully solving one-variable equations is crucial as it forms the basis for understanding more complex mathematical concepts.

Isolation of Variables

The isolation of variables is a critical step in solving equations. It means arranging the equation so that the variable is by itself on one side of the equation. This allows you to clearly see the value that satisfies the equation. In the example with \(-5x + 9 = 8\), we isolated \( x \) by first eliminating the constant \(+ 9\) through subtraction:

- Subtract \( 9 \) from both sides to remove the constant term from the left: \(-5x + 9 - 9 = 8 - 9\)

Doing this leaves us with the equation, \(-5x = -1\). To fully isolate \( x \), we divide both sides by \(-5\):

• Divide each side by the coefficient of the variable, which in this case is \(-5\).
• This results in \( x = \frac{1}{5} \), showing the solution.

Isolation is essential because it simplifies the equation to make the unknown visible and reveals what operation is necessary to complete the solution process.