Problem 27 - Intermediate Algebra

Question

Solve each equation. Check each result. See Example 2. $$ 0.7-4 y=1.74 $$

Step-by-Step Solution

Answer

The solution is \( y = -0.26 \).

Step 1: Isolate the term with the variable

Start with the equation: \[ 0.7 - 4y = 1.74 \]To isolate the term with the variable \( y \), subtract \( 0.7 \) from both sides of the equation. \[ 0.7 - 0.7 - 4y = 1.74 - 0.7 \]This simplifies to: \[ -4y = 1.04 \]

Step 2: Solve for the variable

Now, solve for \( y \) by dividing both sides of the equation by \(-4\): \[ y = \frac{1.04}{-4} \]Calculate the division: \[ y = -0.26 \]

Step 3: Check the solution

Substitute \( y = -0.26 \) back into the original equation to ensure it holds true. Start with the equation: \[ 0.7 - 4(-0.26) = 1.74 \]Calculate \(-4(-0.26)\):\[ = 1.04 \] Then add these results: \[ 0.7 + 1.04 = 1.74 \]The left-hand side equals the right-hand side, verifying that \( y = -0.26 \) is correct.

Key Concepts

Isolating the Variable

When we encounter an equation, our first goal is often to isolate the variable. This means we want the variable, such as \( y \), to be alone on one side of the equation. Why is this important? It allows us to see the value that solves the equation. To isolate the variable, we usually perform operations like addition, subtraction, multiplication, or division.

Let's go through this process using the equation from our example:

• Start with \( 0.7 - 4y = 1.74 \). The term containing the variable, \(-4y\), needs to be isolated.
• Subtract \( 0.7 \) from both sides: \( 0.7 - 0.7 - 4y = 1.74 - 0.7 \).
• This simplifies to \( -4y = 1.04 \). Now, \( y \) is ready to be solved.

Remember, whatever operation you do to one side of the equation, you must do the same to the other to maintain balance.

Solving Equations

Once the variable is isolated, we're ready to solve the equation. This step involves further simplifying until we find the actual value of the variable that satisfies the equation. For linear equations like our example, this typically involves operations that will leave the variable alone on one side.

• In our equation \( -4y = 1.04 \), we need to divide both sides by \(-4\) to solve for \( y \).
• This gives us \( y = \frac{1.04}{-4} \).
• Calculating this division, we find \( y = -0.26 \).

Double-check your calculations at this step, as a small error can lead to an incorrect solution.

Checking Solutions

After finding a solution, confirming its correctness is critical. This step ensures that our calculations were accurate and that the value really satisfies the original equation. Checking involves substituting the value back into the equation and seeing if the equation holds.

Here's how to check:

• Replace \( y \) with \(-0.26\) in the original equation \( 0.7 - 4(-0.26) = 1.74 \).
• Calculate \(-4 \times (-0.26) = 1.04\). Notice the sign change due to multiplying negatives.
• Now, add \( 0.7 + 1.04 \), which results in \( 1.74 \).

The left side equals the right side, confirming our solution \( y = -0.26 \) is indeed correct. Always perform this step to ensure accuracy!