Problem 20 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$ 7(6-y)=-3(y-2) $$

Step-by-Step Solution

Answer

y = 9

Step 1: Distribute

Start by distributing the numbers in the original equation. This gives \(42 - 7y = -3y + 6\)

Step 2: Consolidate Like Terms

Now bring terms involving the same variable on one side and the constants on the other to simplify the equation. This results in \(42 - 6 = 7y - -3y\), which we further simplify to \(36 = 4y\)

Step 3: Solve for y

Lastly, isolate y by dividing every term in the equation by 4, yielding \(y = 36 / 4\), so finally \(y = 9\)

Key Concepts

Distributive Property

Understanding the distributive property is critical when working with algebraic expressions and equations. It's a way to simplify complex problems by spreading a multiplying factor across terms within parentheses. For example, when you encounter an equation such as \(7(6-y)=-3(y-2)\), the very first step is to apply the distributive property:

\(7 \times 6 - 7 \times y = -3 \times y + 3 \times 2\),
which simplifies to \(42 - 7y = -3y + 6\).

This property allows you to multiply a single term by each of the terms inside the parentheses, streamlining the equation for further manipulation. When working with this property, it's crucial to pay attention to the signs (positive or negative) of the numbers involved to ensure accurate distribution.

Consolidating Like Terms

Once the distributive property has been applied, the next step often involves consolidating like terms. Like terms are terms that have the same variable raised to the same power. In our equation, \(42 - 7y = -3y + 6\), we identify \(7y\) and \(3y\) as like terms. Consolidating means to combine these terms to simplify the equation further.

By bringing the like terms \(7y\) and \(3y\) together to one side, and the constants, 42 and 6, to the other side, we get:
\(42 - 6 = 7y - (-3y)\),
which simplifies to \(36 = 4y\).

Here, consolidating like terms makes the equation more manageable and sets the stage for the final step of solving for the variable.

Isolating Variables

To find the solution to linear equations like the one we're working with, isolating the variable is the final stage. Having simplified our example to \(36 = 4y\), we need to get \(y\) on its own on one side of the equation. We do this by performing the same mathematical operation on both sides of the equation, which in this case is division by 4:

\(y = \frac{36}{4}\).

Once you've divided both sides by 4, the variable \(y\) is isolated, and you're left with \(y = 9\), which is the solution to the equation. Isolating the variable involves reversing the operations that are being applied to the variable. If the variable is being multiplied, you'll divide, and if it's being added to, you'll subtract. Mastering this skill is key to solving many types of algebra problems.