Problem 2 - Algebra Form and Function

Question

Solve the equations. $$ \frac{5}{3}(y+4)=\frac{1}{2}-y $$

Step-by-Step Solution

Answer

Answer: The value of y in the given equation is $y = \frac{-37}{13}$.

Step 1: Remove Fractions by Multiplying Both Sides by the Least Common Multiple

In order to eliminate the fractions, we should find the least common multiple (LCM) of the denominators (3 and 2). The LCM for 3 and 2 is 6. Multiply both sides of the equation by 6. $$6\left(\frac{5}{3}(y+4)\right) = 6\left(\frac{1}{2}-y\right)$$

Step 2: Simplify the Fractions

Now we will simplify the fractions on both sides of the equation. $$\frac{6}{3}\cdot 5(y+4) = \frac{6}{2}\cdot (1-y)$$ $$2\cdot 5(y+4) = 3\cdot (1-y)$$ $$10(y+4) = 3(1-y)$$

Step 3: Distribute the Multiplication

We will distribute the multiplication of 10 and 3 on both sides of the equation. $$10y + 40 = 3 - 3y$$

Step 4: Move all y Terms to One Side and All Constant Terms to the Other Side

Now, we will move all the y terms to one side of the equation (left side) and all the constant terms to the other side (right side). $$10y + 3y = 3 - 40$$

Step 5: Combine Like Terms

Combine like terms: $$13y = -37$$

Step 6: Solve for y

Now we can solve for y by dividing by 13: $$y = \frac{-37}{13}$$ Thus, the solution to the given equation is: $$y = \frac{-37}{13}$$

Key Concepts

Least Common Multiple

When solving equations with fractions, finding the least common multiple (LCM) of the denominators helps to eliminate the fractions. This step simplifies the equation, making it easier to solve.
To find the LCM of two numbers, identify the smallest number that both can divide evenly. In the given equation, the denominators are 3 and 2.

• List the multiples of the numbers: 3: 3, 6, 9, ... and 2: 2, 4, 6, ...
• Identify the smallest multiple common to both lists. Here, 6 is the smallest shared multiple.

Once the LCM, 6, is determined, multiply every term by 6 to remove the fractions. Scaling up this way maintains the equation’s balance. It allows you to work with whole numbers, which simplifies further manipulation and calculations.

Combining Like Terms

In the process of solving equations, combining like terms is an essential skill to simplify expressions. Terms are 'like' when they involve the same variable raised to the same power. This allows you to merge them into a single term.
For example, in the equation after distribution, you have 10y on one side and -3y on the other.

• Move the y terms to one side to simplify: 10y + 3y
• Combine them to get: 13y

Similarly, combining constant numbers without variables can simplify further steps of finding unknown values, like moving 3 and 40 to the right side.
This method condenses the equation, reducing it to a more manageable expression that allows straightforward resolution.

Distributing Multiplication

Distributing multiplication involves multiplying each term inside a parenthesis by a number outside of it. This step is crucial to simplify and solve linear equations effectively.
In the previous equation, once the fractions are removed, you'll distribute the multiplied amounts:

• For the left side: 10(y + 4), distribute gives 10y + 40
• For the right side: 3(1 - y), distribute gives 3 - 3y

This step ensures each term inside the parentheses experiences the outside factor. Proper distribution lays the foundation for further simplification like moving terms and combining them.
Distributing correctly is vital to maintaining the equation's integrity, allowing you to uncover the variable step-by-step.