Problem 40 - Algebra 1

Question

Solve the equation $$ 5 x-(6-x)=2(x-7) $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = -2 \)

Step 1: Simplify the equation

Start by simplifying the left-hand side of the equation. The expression 5x - (6 - x) can be simplified by distributing the negative sign over the terms within the brackets to get 5x - 6 + x. After simplifying, the left side of the equation becomes 6x - 6.

Step 2: Simplify the right-hand side of the equation

The right-hand side of the equation is 2(x - 7). Distribute 2 over (x - 7) to get 2x - 14.

Step 3: Write the simplified equation

After simplifying both sides, the equation now reads 6x - 6 = 2x - 14.

Step 4: Solve for x

To solve for x, start by moving like terms together. Subtract 2x from both sides to get 4x - 6 = -14. Then add 6 to both sides to get 4x = -8. Divide both sides by 4 to isolate x, which gives x = -2.

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of numbers, variables (like \( x \)), and mathematical operators (such as addition, subtraction, multiplication, and division). They are the fundamental building blocks of algebra and essential for forming equations. In the given problem, expressions like \( 5x \), \(- (6-x) \), and \( 2(x-7) \) are examples of algebraic expressions.
Algebraic expressions allow us to model real-world problems with mathematical statements that can be manipulated. Understanding how to work with these is key to solving equations. When combining like terms in an expression, you simplify it by adding or subtracting coefficients of variables that are the same. Let's look at an example:

• Combine \( 5x \) and \( -x \) to get \( 4x \).
• Add or subtract constants, like \( 6 \) and \(- 2 \), to simplify further.

This helps in streamlining the equation to make solving simpler. Remember, the goal is to turn complex expressions into simpler ones for easy problem-solving.

Distributive Property

The distributive property is a key concept that allows you to multiply a single term by terms within a bracket or parentheses. It is expressed as \( a(b + c) = ab + ac \). This property is pivotal when working with algebraic expressions and solving equations.
In the given problem, using the distributive property involves multiplying each term inside the parentheses by \( -1 \) in \( -(6-x) \), and \( 2 \) in \( 2(x-7) \).

• For \( -(6-x) \), distribute the negation: \(-1 \cdot 6 + (-1) \cdot (-x) = -6 + x \).
• For \( 2(x-7) \), distribute the \( 2 \): \( 2 \cdot x + 2 \cdot (-7) = 2x - 14 \).

Understanding how to apply this property correctly will significantly simplify complex expressions and allow you to manipulate equations more effectively. It is crucial to always perform distribution carefully to avoid errors in calculation.

Linear Equations

Linear equations are equations that make a straight line when graphed. They often appear in the form \( ax + b = 0 \), where \( a \) and \( b \) are constants, and \( x \) is a variable. Solving linear equations is all about finding the value of the variable that makes the equation true.
In this exercise, after simplifying both sides, we have the linear equation \( 6x - 6 = 2x - 14 \). The steps to solve it included:

• Moving variables to one side: by subtracting \( 2x \) from both sides, we align variables toward the left side, resulting in \( 4x - 6 = -14 \).
• Isolating the variable: by adding \( 6 \) to both sides to remove constant terms, we achieve \( 4x = -8 \).
• Solving for the variable: finally, dividing by \( 4 \) gives \( x = -2 \).

Solving linear equations involves manipulating both sides of the equation to isolate the variable, making it straightforward to find its value. By mastering these steps, you can tackle a wide range of linear equations with confidence.