Problem 50 - Precalculus Student Solutions Manual 5th

Question

Solve each equation for \(x\). $$5 x-(2 a+c)=4(x+c)$$

Step-by-Step Solution

Answer

The solution for x is \( x = 2a + 5c \).

Step 1 - Distribute terms on the right side

Distribute the 4 on the right side of the equation: \[ 5x - (2a + c) = 4x + 4c \]

Step 2 - Move all terms involving x to one side

Subtract 4x from both sides of the equation to isolate the terms involving x on one side: \[ 5x - 4x - (2a + c) = 4c \]This simplifies to: \[ x - (2a + c) = 4c \]

Step 3 - Move constant terms to the other side

Add (2a + c) to both sides to isolate x: \[ x = 4c + 2a + c \]Combine like terms on the right side: \[ x = 2a + 5c \]

Key Concepts

linear equations

Linear equations are fundamental in algebra. They are equations of the first order, meaning they involve variables raised to the power of one. These equations take the form ax + b = c , where 'a,' 'b,' and 'c' are constants, and 'x' is the variable. The goal in solving linear equations is to find the value of 'x' that makes the equation true. Let's explore simple steps and techniques to solve linear equations.

algebraic manipulation

Algebraic manipulation involves rearranging equations to simplify and solve for a variable. Here are some key techniques:

• Adding or subtracting the same value from both sides of the equation.
• Multiplying or dividing both sides of the equation by the same nonzero value.

These operations do not change the equality, but they help in isolating the variable. Practicing these tactics will make solving equations much easier.

distributive property

The distributive property is an important algebraic property used to simplify expressions. It states that: a(b + c) = ab + ac . In our given equation 5 x-(2 a+c)=4(x+c) , applying the distributive property helps us deal with parentheses. For example, 4(x+c) becomes 4x + 4c when distributed. Distributing terms efficiently can clarify complex equations and make them more manageable.

isolating variables

Isolating the variable means getting the variable (x) alone on one side of the equation. This is typically done in stages:

• Move all terms containing the variable to one side of the equation.
• Move all constant terms to the opposite side.

For example, in the equation x - (2a + c) = 4c , we isolate 'x' by adding (2a + c) to both sides, which results in x = 4c + 2a + c . Combining like terms leads us to the final solution: x = 2a + 5c .