Problem 55 - Elementary Algebra

Question

Solve. $$ 3-2(x+4)=-3(4 x-5) $$

Step-by-Step Solution

Answer

The solution is \( x = 2 \).

Step 1: Expand Expressions

Let's first expand the expressions inside each parenthesis. Distribute the variables:- On the left-hand side: \[-2(x+4) = -2x - 8\] So, replace \(-2(x+4)\) with \(-2x - 8\).- On the right-hand side: \[-3(4x-5) = -12x + 15\] Replace \(-3(4x-5)\) with \(-12x + 15\).The equation becomes:\[3 - 2x - 8 = -12x + 15\]

Step 2: Combine Like Terms on the Left

Combine like terms on the left side of the equation. Here, combine the constants 3 and -8:\[3 - 8 = -5\]Therefore, the equation now is:\[-5 - 2x = -12x + 15\]

Step 3: Isolate Variables

Add \(12x\) to both sides to get variables on one side:\[-5 - 2x + 12x = 15\]This simplifies to:\[-5 + 10x = 15\]

Step 4: Solve for x

Add 5 to both sides to isolate terms with \(x\):\[-5 + 10x + 5 = 15 + 5\]This simplifies to:\[10x = 20\]Then, divide both sides by 10 to solve for \(x\):\[x = \frac{20}{10} = 2\]

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that allows you to simplify expressions where a single term is multiplied by terms inside a parenthesis. The rule states that

• a(b + c) = ab + ac
• a(b - c) = ab - ac

In this exercise, we used the distributive property to expand the expression

• On the left side: e.g., \(-2(x+4) = -2x - 8\)
• On the right side: e.g., \(-3(4x-5) = -12x + 15\)

This step is crucial as it removes the parentheses, allowing us to simplify the equation further. Distributing is like taking each term inside the parenthesis and multiplying it by the term outside.

Combining Like Terms

Combining like terms is an essential step in simplifying expressions and equations. "Like terms" are terms that have the same variable raised to the same power.
For instance, in the equation obtained after distribution
: \[3 - 2x - 8 = -12x + 15\]
, we need to combine constants together.

• Combine the constants on the left: \(3 - 8\) becomes \(-5\)

After combining, the equation looks cleaner and easier to work with: \[-5 - 2x = -12x + 15\]Combining like terms is a straightforward, yet powerful, technique to streamline and organize your equation, paving the way for isolating variables.

Solving for Variables

Solving for variables involves manipulating the equation to isolate the variable on one side, often making it ready for mathematical evaluation. Our goal is to get the variable by itself. Here’s how we progress in our example:

• Move all terms involving the variable \(x\) to one side. Add \(12x\) to both sides: \[-5 - 2x + 12x = 15\]which simplifies to \[-5 + 10x = 15\]

Solving equations involves step-by-step adjustments to isolate terms with \(x\), often ending in a simpler operation to reveal the variable's value. This part of the process teaches us how algebra can organize and simplify the problem to reach a solution.

Linear Equations

Linear equations are the simplest type of algebraic equations, where each term is either a constant or the product of a constant and a single variable. These equations are linear because they form a straight line when graphed. Our main target is to find the value of the unknown variable. In our example, the equation becomes:
\[10x = 20\]
After simplifying and isolating the terms with the variable,

• Divide both sides by \(10\): \[x = \frac{20}{10} = 2\]

Linear equations are foundational in algebra because they set the stage for understanding more complex equations and systems of equations. They also reinforce essential principles like balancing both sides of the equation. With linear equations, we often aim for simplicity and clarity.