Problem 46 - Algebra 1

Question

Solve the equation. Round the result to the nearest tenth if necessary. $$-1.3 y+3.7=4.2-5.4 y$$

Step-by-Step Solution

Answer

The solution to the equation is \(y = 0.5 / 4.1\). Remember to round to the nearest tenth if necessary.

Step 1: Combine like terms

Add \(5.4y\) to both sides of the equation to combine terms with y. This results in \( (-1.3y + 5.4y) + 3.7 = 4.2 \), which simplifies to \(4.1y + 3.7 = 4.2\).

Step 2: Isolate y

To isolate y, subtract \(3.7\) from both sides of the equation. You then get \(4.1y = 4.2 - 3.7\), which is \(4.1y = 0.5\).

Step 3: Solve for y

Finally, to solve for y, you have to divide both sides of the equation by \(4.1\). This gives us \(y = 0.5 / 4.1\).

Key Concepts

Combining Like Terms

When solving equations, combining like terms simplifies the expression. Like terms are terms that contain the same variable raised to the same power. In our equation

• \(-1.3y + 3.7 = 4.2 - 5.4y\)

we have terms with the variable \(y\) on both sides.

To combine them, add \(5.4y\) to both sides. This cancels out the \(-5.4y\) on the right side, helping us group all \(y\) terms together. It transforms the equation to

• \(( -1.3y + 5.4y ) + 3.7 = 4.2\)

Simplifying gives us \(4.1y + 3.7 = 4.2\). By grouping terms in this way, we create a simpler equation that's easier to handle.

Always look for like terms before trying to solve the equation.

Isolating Variables

To solve an equation, our goal is to find the value of the variable. Isolating the variable makes this possible. In the equation we are working with

• \(4.1y + 3.7 = 4.2\)

you need \(y\) by itself on one side.

To isolate \(y\), subtract \(3.7\) from both sides:

• \(4.1y = 4.2 - 3.7\)

This subtraction leaves us with \(4.1y = 0.5\).

Always perform the same operation on both sides to maintain the equation's balance. This step is crucial to ensuring each side remains equal as you simplify the equation.

Dividing to Solve

Once the variable is isolated, the next step is to solve for it by dividing. After isolating \(y\), you ended up with:

• \(4.1y = 0.5\)

Here, \(y\) is multiplied by \(4.1\), so divide both sides by \(4.1\):

• \(y = \frac{0.5}{4.1}\)

This gives us the value of \(y\). Use a calculator to find that \(y \approx 0.12\), rounded to the nearest tenth.

Dividing lets us solve for the variable, completing the equation's solution. Always check your work by substituting back into the original equation to ensure it's correct.