Problem 28 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$ 10 y=2 y-6 y+7 $$

Step-by-Step Solution

Answer

The solution to the equation is \(y = 0.5\).

Step 1: Combining Like Terms

Simplify the left-hand side (LHS) and the right-hand side (RHS) of the equation. In the RHS of given equation, \(2y\) and \(-6y\) are like terms, therefore they can be combined. So, the equation becomes \(10y = -4y + 7\).

Step 2: Bring variables to one side

To isolate \(y\), get all the terms with \(y\) on one side by adding \(4y\) to both sides. This will give: \(10y + 4y = -4y + 4y + 7\) which simplifies to \(14y = 7\).

Step 3: Solve for y

Now, to isolate \(y\), divide both sides of the equation by 14. So the equation becomes \(y = 7/14\). Simplify the RHS to get the final value of \(y\).

Key Concepts

Combining Like Terms

When solving linear equations, combining like terms is a crucial first step. Like terms are simply terms that contain the same variable raised to the same power. They can be added or subtracted from one another. In equation solving, combining like terms simplifies the expression, making it easier to solve.

Consider the expression on the right-hand side of the equation provided: \(2y - 6y + 7\). Here, both \(2y\) and \(-6y\) are like terms because they both contain the variable \(y\). By combining these like terms, you perform the following calculation:

• \(2y - 6y = -4y\)

Now, the equation is simplified to \(10y = -4y + 7\). Notice how the equation becomes less cluttered, bringing you a step closer to isolating the variable.

Isolating Variables

After simplifying the equation by combining like terms, the next step is to isolate the variable. Isolating a variable means manipulating the equation such that the variable appears on one side while constants appear on the other. This helps in solving for the variable efficiently.

To isolate \(y\) in our current equation \(10y = -4y + 7\), bring all terms containing \(y\) to one side of the equation. You do this by adding \(4y\) to both sides:

• \(10y + 4y = -4y + 4y + 7\)
• Simplifies to: \(14y = 7\)

By doing this, all terms with \(y\) have been consolidated, leaving the equation with \(y\) on one side and a constant on the other, thereby isolating \(y\).

Simplifying Fractions

Once the variable is isolated on one side, often it results in a fractional equation. Simplifying the fraction is the final step. Simplification involves reducing the fraction to its lowest terms. This makes it easier to understand and use the result in further calculations.

In our example, we reached \(14y = 7\). To solve for \(y\), divide both sides by 14:

• \(y = \frac{7}{14}\)

The fraction \(\frac{7}{14}\) simplifies further by identifying and dividing by the greatest common divisor, which is 7:

• \(\frac{7}{14} = \frac{1}{2}\)

Therefore, the value of \(y\) is \(\frac{1}{2}\). Always ensure fractions are in their simplest form for clarity and ease of use.