Problem 28 - Prealgebra

Question

Solve equation. Check your solution. \(12.4 y+14=6 y-2\)

Step-by-Step Solution

Answer

The solution is \(y = -2.5\).

Step 1: Isolate the variable terms

Start by getting all the terms with the variable on one side of the equation. Subtract \(6y\) from both sides of the equation: \[12.4y - 6y + 14 = 6y - 6y - 2\] This simplifies to: \[6.4y + 14 = -2\]

Step 2: Move constant terms

Next, isolate the term with the variable by subtracting \(14\) from both sides: \[6.4y + 14 - 14 = -2 - 14\] This yields: \[6.4y = -16\]

Step 3: Solve for the variable

Now, solve for \(y\) by dividing both sides by \(6.4\): \[y = \frac{-16}{6.4}\] Calculate the division: \[y = -2.5\]

Step 4: Check the solution

Substitute \(y = -2.5\) back into the original equation to verify: Start with the original equation: \[12.4y + 14 = 6y - 2\] Substitute \(y\): \[12.4(-2.5) + 14 = 6(-2.5) - 2\] Calculate both sides: Left side: \(12.4 imes -2.5 + 14 = -31 + 14 = -17\) Right side: \(6 imes -2.5 - 2 = -15 - 2 = -17\) Since both sides equal, the solution is correct.

Key Concepts

Variable Isolation

In any equation, one primary goal is to solve for the unknown variable. To start, we need to gather all the terms involving the variable onto one side of the equation. This process is referred to as "variable isolation." It helps to simplify the equation so you can clearly see what needs to be done next.

In the equation \(12.4y + 14 = 6y - 2\), our variable is \(y\). We want all \(y\) terms on one side, normally the left. By subtracting \(6y\) from both sides, we ensure that all the "\(y\)" stuff remains on the same side. This action gives us \(12.4y - 6y + 14 = -2\), which simplifies to \(6.4y + 14 = -2\).

These steps are crucial because once the variable is isolated on one side, we can concentrate on getting rid of other terms (like constants) to solve the equation.

Constant Term

The constant term in an equation is a fixed value and doesn't include any variables. When resolving an equation, handling constant terms spent next after isolating variable terms. In our equation \(6.4y + 14 = -2\), the number \(14\) is a constant term that we want to move away from the \(y\).

To "get rid" of or move a constant term, you use the opposite operation. Here, the constant term "\(+14\)" requires us to subtract \(14\) from both sides to maintain the equation's balance. Doing this results in \(6.4y = -16\).

This way of moving constant terms clears the path for the variable "\(y\)", making further steps like division much simpler.

Division

After isolating the variable and removing constants, the next step is often to solve for the variable itself by making its coefficient equal to one. This often involves using division.

In our equation \(6.4y = -16\), \(6.4\) is the coefficient of \(y\). To solve for \(y\), divide both sides by \(6.4\). This is because any number divided by itself equals one, simplifying \(6.4y \div 6.4\) to just \(y\). The equation then becomes \(y = -16 \div 6.4\).

Calculating this division gives \(y = -2.5\). Always confirm your solution is correct by substituting \(y\) back into the original equation to ensure both sides of the equation remain equal, as this verifies the accuracy of your division and other operations.