Problem 53 - Algebra A Combined Function

Question

Solve each equation. See Section \(2.3 .\) \(5(x-3)+3 x=1\)

Step-by-Step Solution

Answer

The solution is \(x = 2\).

Step 1: Distribute the 5

First, apply the distributive property to expand the expression on the left-hand side of the equation. Multiply 5 by both terms inside the parentheses: \[ 5(x - 3) + 3x = 1 \] Becomes: \[ 5x - 15 + 3x = 1 \]

Step 2: Combine like terms

Next, combine the like terms on the left-hand side of the equation. The terms \(5x\) and \(3x\) are like terms. Add them together:\[ 8x - 15 = 1 \]

Step 3: Isolate the variable term

Add 15 to both sides of the equation to isolate the term with the variable on one side:\[ 8x - 15 + 15 = 1 + 15 \] This simplifies to: \[ 8x = 16 \]

Step 4: Solve for x

Finally, divide both sides by 8 to solve for \(x\):\[ \frac{8x}{8} = \frac{16}{8} \] This reduces to: \[ x = 2 \]

Key Concepts

Distributive Property

The distributive property is an essential concept in solving linear equations, especially when dealing with parentheses.When you see an expression like \(5(x - 3)\), you need to multiply the 5 by each term inside the parentheses. In this case, that means distributing 5 to both \(x\) and -3.

• Multiply 5 by \(x\) to get \(5x\).
• Multiply 5 by -3 to get -15.

This process transforms the expression into \(5x - 15\).
Distributing helps simplify equations, making it easier to work with individual terms.

Combining Like Terms

Once you've distributed, the next step is to combine like terms. Like terms are terms that have the same variable raised to the same power.In the equation \(5x - 15 + 3x = 1\), \(5x\) and \(3x\) are like terms.

• Identify terms with the same variable, like \(5x\) and \(3x\).
• Add their coefficients: \(5 + 3 = 8\).

After combining, the equation simplifies to \(8x - 15 = 1\).
Combining like terms helps reduce the number of terms, making the equation easier to solve.

Isolating the Variable

Isolating the variable is a crucial step in solving any equation. The goal is to get the variable by itself on one side of the equation.In the equation \(8x - 15 = 1\), you want \(x\) alone.
To do this, perform operations that maintain the equation's balance.

• Add 15 to both sides: \(8x - 15 + 15 = 1 + 15\).
• This simplifies to \(8x = 16\).

Every operation done to one side must be done to the other side to keep everything equal.
Isolating the variable sets you up perfectly for the final step.

Algebraic Manipulation

Algebraic manipulation ties all the previous steps together to solve the equation.Once the variable is isolated, you're ready for final operations.In \(8x = 16\), divide both sides by 8 to solve for \(x\).

• Divide: \(\frac{8x}{8} = \frac{16}{8}\).
• This simplifies to \(x = 2\).

Algebraic manipulation involves performing arithmetic operations like addition, subtraction, multiplication, and division to both sides until you find the value of the variable.Consistency in these operations ensures the equation remains balanced throughout.