Problem 10 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. $$7 x-2(x-2)=12$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 1.6\)

Step 1: Apply distributive property

Firstly, apply the distributive property to simplify the equation. In \[7x - 2(x - 2) = 12\], distribute \(-2\) across \(x - 2\) to get: \[7x - 2x + 4 = 12\]

Step 2: Combine like terms

Combine like terms on the left side of the equation: \[ (7x - 2x) + 4 = 12\] which simplifies to \[5x + 4 = 12\]

Step 3: Isolate the variable

Subtract 4 from both sides of the equation to isolate \(5x\): \[5x + 4 - 4 = 12 - 4\] which simplifies to \[5x = 8\]. Finally divide both sides of the equation by 5 to solve for x: \[x = 8 / 5\] or \[x = 1.6\]

Key Concepts

Distributive Property

The distributive property is a crucial algebraic rule used for simplifying mathematical expressions. It states that for any numbers a, b, and c, the equation a(b + c) is equal to ab + ac. In other words, a single term outside the parenthesis can be distributed, or multiplied, with each term inside the parenthesis.

Let's see how this applies to our exercise. We have an expression \(7x - 2(x - 2) = 12\). Before we can solve for \(x\), we need to get rid of the parenthesis. We apply the distributive property by multiplying -2 with both \(x\) and -2:

• \( -2 \times x \) becomes \(-2x\)
• \(-2 \times (-2)\) becomes \( +4 \)

This gives us a simplified equation without parenthesis: \(7x - 2x + 4 = 12\). Understanding the distributive property is essential in algebra to simplify expressions and solve equations efficiently.

Combining Like Terms

Combining like terms is a technique for simplifying algebraic expressions or equations. Terms that have the same variables raised to the same power are 'like terms,' and they can be combined by adding or subtracting their coefficients. This process is crucial as it reduces the complexity of equations, allowing for easier manipulation and solution.

Looking at our simplified equation \(7x - 2x + 4 = 12\), we notice the terms \(7x\) and \(-2x\) are like terms. To combine them, we just add their coefficients (7 and -2) together while keeping the variable part unchanged:

• \(7 - 2 = 5\)

Thus, \(7x - 2x\) simplifies to \(5x\). Now our equation looks like \(5x + 4 = 12\), which is simpler and brings us one step closer to solving for \(x\).

Isolating Variables

Isolating variables is a fundamental step in solving linear equations. To find the value of the unknown variable, you must manipulate the equation to get the variable alone on one side of the equation. The aim is to have the variable term by itself, equaling a number or a simpler expression.

In our exercise, after combining like terms, we have the equation \(5x + 4 = 12\). To isolate \(x\), we need to perform operations that will eliminate other terms:

• First we subtract 4 from both sides to remove the number alongside the variable term: \(5x + 4 - 4 = 12 - 4\), which simplifies to \(5x = 8\).
• Then, we divide both sides by the coefficient of \(x\), which is 5, to get the variable alone: \(\frac{5x}{5} = \frac{8}{5}\), resulting in \(x = 1.6\).

This process of isolation is the key to solving for the variable and finding the solution to the equation.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations (like addition, subtraction, multiplication, and division) without an equals sign. They are the building blocks of algebra and are used to represent values in mathematical problems.

In our example, the initial equation \(7x - 2(x - 2) = 12\) contains an algebraic expression on the left-hand side. An expression can be simplified using the above mentioned properties - the distributive property and combining like terms. Once the expression is simplified, we can proceed to solve the equation by isolating the variable. Algebraic expressions must be handled with an understanding of these properties to ensure accurate simplification and solution of equations.