Problem 85 - Algebra A Combined Function

Question

Use a calculator to determine the solution of each equation. $$ 36.766+x=-108.712 $$

Step-by-Step Solution

Answer

The solution is \( x = -145.478 \).

Step 1: Understand the Equation

Take a look at the equation: \( 36.766 + x = -108.712 \). The problem asks you to find the value of \( x \) that makes this equation true.

Step 2: Isolate x on One Side

To solve for \( x \), subtract \( 36.766 \) from both sides of the equation to isolate \( x \). This gives:\[ x = -108.712 - 36.766 \]

Step 3: Perform the Calculation

Use your calculator to compute the value of \( -108.712 - 36.766 \). The calculation results in \( x = -145.478 \).

Step 4: Verify Solution

To check your work, substitute \( x = -145.478 \) back into the original equation:\[ 36.766 + (-145.478) = -108.712 \]This confirms that the original equation is true when \( x = -145.478 \).

Key Concepts

Isolation of Variables

When solving equations, one common goal is to isolate the variable you want to find—in this case, the variable is \( x \). Isolating \( x \) means that you want it to stand alone on one side of the equation. This process simplifies the equation and makes it easy to solve.

• Identify the variable to isolate; here, it is \( x \).
• Perform operations that will leave the variable by itself on one side of the equation.

This is often done by adding, subtracting, multiplying, or dividing both sides of the equation by the same number. In this example, subtracting 36.766 from both sides isolates \( x \). Once isolated, you have a clean equation to solve.

Subtraction

Subtraction is a key mathematical operation used to manipulate equations. It helps in isolating variables, just like in our example problem where we had to subtract \( 36.766 \) from both sides to solve for \( x \).Subtracting the same number from both sides doesn't change the balance of the equation. Think of an equation like a balance scale: each side should weigh the same. When you subtract from one side, do the same for the other.

• Start by identifying the number you need to subtract to isolate the variable.
• Remember, whatever you do to one side, do equally to the other side.

Using subtraction keeps equations balanced and helps in simplifying them to a more solvable form.

Calculator Usage

Using a calculator can make solving equations much easier, especially when dealing with decimals or complex numbers. Calculators can quickly handle operations like addition, subtraction, multiplication, and division. For many, it's an essential tool in math homework.

• Enter the numbers and operations exactly as they appear in the equation.
• Verify you have transcribed the numbers correctly to prevent errors.
• Press Enter or Equals to get your result.

In our exercise, input \( -108.712 - 36.766 \) into the calculator to find that \( x = -145.478 \). Always double-check your input on the calculator to ensure accuracy.

Verification of Solutions

Verifying solutions is a crucial step in confirming the correctness of your answer. Once you have an answer, plug it back into the original equation to see if both sides equal.

• Substitute the value of \( x \) in the original equation.
• Perform the operations on both sides.
• Check if the left side equals the right side of the equation.

In our example, substitute \( -145.478 \) back in place of \( x \) in the original equation \( 36.766 + x = -108.712 \). When both sides equal, it confirms that \( x = -145.478 \) is indeed the correct solution.