Problem 12 - Intermediate Algebra

Question

a. Simplify: \(5 y+2-3 y\) b. Solve: \(5 y+2-3 y=8\) c. Evaluate \(5 y+2-3 y\) for \(y=8\) d. Check: Is \(-1\) a solution of \(5 y+2-3 y=8 ?\)

Step-by-Step Solution

Answer

a. Simplified: \(2y + 2\). b. \(y = 3\). c. Evaluated: 18. d. \(-1\) is not a solution.

Step 1: Simplify the expression

Start with the expression \(5y + 2 - 3y\). Combine like terms by subtracting the coefficients of \(y\): \(5y - 3y = 2y\). Thus, the expression simplifies to \(2y + 2\).

Step 2: Solve the equation

Given the equation \(5y + 2 - 3y = 8\). Using the simplified expression from Step 1, this becomes \(2y + 2 = 8\). Subtract 2 from both sides to isolate the term with \(y\): \(2y = 6\). Divide both sides by 2 to solve for \(y\): \(y = 3\).

Step 3: Evaluate the expression for \(y = 8\)

Using the simplified expression \(2y + 2\), plug in \(y = 8\): \(2(8) + 2 = 16 + 2 = 18\). Thus, the expression evaluates to 18 when \(y = 8\).

Step 4: Check if \(-1\) is a solution to the equation

Substitute \(y = -1\) into the equation \(5y + 2 - 3y = 8\). Simplify: \(5(-1) + 2 - 3(-1) = -5 + 2 + 3 = 0\). Since \(0 eq 8\), \(-1\) is not a solution of the equation.

Key Concepts

Simplifying Expressions

Simplifying algebraic expressions is like cleaning your room. Just like you put similar items together and remove any unnecessary clutter, simplifying an expression means combining like terms. In the expression \(5y + 2 - 3y\), "like terms" are those that involve the same variable raised to the same power. Here, both \(5y\) and \(-3y\) are like terms because they involve the variable \(y\). To simplify, we subtract the coefficients of \(y\):

• Start with the term \(5y\).
• Subtract \(3y\) from \(5y\), which gives \(2y\).

So, the expression becomes \(2y + 2\). The constant term \(2\) remains unchanged as there is no other constant term to combine it with. Recognizing and organizing like terms can make further steps, like solving equations, easier!

Solving Linear Equations

Solving linear equations finds the value of the variable that makes the equation true. Take the simplified form of the previous expression: \(2y + 2 = 8\). Our goal is to find \(y\). This involves reversing the operations applied to \(y\).

• First, isolate the \(y\)-term by subtracting \(2\) from both sides of the equation:

\[2y + 2 - 2 = 8 - 2\]This simplifies to \(2y = 6\).

• Next, divide both sides by \(2\) to solve for \(y\):

\[\frac{2y}{2} = \frac{6}{2}\]So, \(y = 3\). Each step ensures the equation remains balanced, much like keeping both sides of a scale even.

Evaluating Expressions

Evaluating an expression involves substituting a specific value for the variable and calculating to find a final numeric result. Let's consider the simplified expression \(2y + 2\) and evaluate it for \(y = 8\).

• Substitute \(8\) in place of \(y\):

\[2 imes 8 + 2\]

• First, multiply \(2\) by \(8\):

\[16 + 2\]

• Then, add the \(2\) to get \(18\).

This result tells us the value of the expression when \(y\) is specifically \(8\). Evaluating expressions is crucial for determining outputs based on different inputs.

Checking Solutions

Checking solutions is the process of verifying that a proposed solution indeed satisfies the original equation. Let's see if \(-1\) is a solution to the equation \(5y + 2 - 3y = 8\).

• Substitute \(-1\) for \(y\) in the equation:

\[5(-1) + 2 - 3(-1)\]

• Simplify by performing the operations:

\[-5 + 2 + 3\]

• You end up with \(0\).

Since \(0 eq 8\), \(-1\) is not a valid solution. Checking solutions helps confirm whether the value truly solves the equation, ensuring accuracy in problem-solving.