Question

Solve each of the given equations for \(x .\) Check your answers by hand or with a calculator $$0.5 x+9=4.5 x+17$$

Short answer

The solution is \( x = -2 \).

Step-by-step solution

Isolate the variable

Start by getting all terms with the variable on one side of the equation. Subtract \( 0.5x \) from both sides: \( 0.5x + 9 - 0.5x = 4.5x + 17 - 0.5x \), resulting in \( 9 = 4x + 17 \).

Simplify the equation

Subtract 17 from both sides to get the terms involving \( x \) on one side and constants on the other. This gives: \( 9 - 17 = 4x \) which simplifies to \( -8 = 4x \).

Solve for x

To isolate \( x \), divide both sides of the equation by 4: \( \frac{-8}{4} = \frac{4x}{4} \). Thus, \( x = -2 \).

Check the solution

Substitute \( x = -2 \) back into the original equation to ensure it is correct: \( 0.5(-2) + 9 = 4.5(-2) + 17 \). Simplify both sides: \( -1 + 9 = -9 + 17 \), which gives \( 8 = 8 \). Since both sides are equal, the solution is verified.

Key concepts

Isolation of Variables

The first step in solving an equation is to isolate the variable you are solving for, in this case, \(x\). Isolation of variables means that you want to get the variable on one side of the equation by itself. This involves manipulating the equation so that all the terms with \(x\) are on one side and the constants are on the other.

Here's how it's done:

• Identify all the terms that include the variable. In this example, the terms are \(0.5x\) and \(4.5x\).
• Decide which side you want the variable to be on. Typically, you'll move the term with the smaller coefficient, which in this case is \(0.5x\), to keep calculations simpler.
• Subtract \(0.5x\) from both sides to eliminate it from the left side of the equation: \(0.5x + 9 - 0.5x = 4.5x + 17 - 0.5x\). This step results in the equation \(9 = 4x + 17\), effectively isolating the variable terms on the right side.
• Continue with further steps to fully isolate \(x\)."

Remember, isolating the variable is a crucial step that sets the stage for simplifying the equation and finding the solution.

Equation Simplification

Once we have the variable isolated on one side, the next focus is to simplify the equation. Simplification is making the equation as straightforward as possible, which helps to easily solve for the variable.

Here's what happens:

• After isolating the variable terms, we subtract the constant from both sides to balance the equation. In the example, subtract 17 from both sides to get: \(9 - 17 = 4x\), which simplifies to \(-8 = 4x\).
• The goal here is to narrow down the number of terms on each side so that solving for \(x\) becomes a clear and easy process.
• With the equation \(-8 = 4x\), the coefficient of \(x\) is greater than 1, so we proceed to simplify it further.
• This involves dividing both sides by the coefficient of \(x\), in this case 4, to isolate \(x\) completely: \(\frac{-8}{4} = x\), which leads us to the solution \(x = -2\).

By simplifying the equation, you transform a complicated equation into a simple one, making it easier to interpret and find the solution.

Checking Solutions

Once you have found the solution for \(x\), it is very important to check if your solution is correct. Checking solutions is the process of substituting the found value back into the original equation to ensure that it truly satisfies the equation.

Here's how to perform a solution check:

• Take the value of \(x\) you found, in this case \(x = -2\), and substitute it back into the original equation \(0.5x + 9 = 4.5x + 17\).
• Calculate each side separately: Substitute \(-2\) into the left side to get \(0.5(-2) + 9 = -1 + 9 = 8\). For the right side, substitute \(-2\) to get \(4.5(-2) + 17 = -9 + 17 = 8\).
• Compare both sides of the equation. If they are equal, then the solution is verified and correct. In our example, both sides equal 8, confirming that \(x = -2\) is indeed the correct solution.

Checking your solution not only guarantees that you have not made any calculation errors but also reinforces your understanding of the solution process.