Problem 48 - Elementary Algebra

Question

Solve. $$ 5 x-3(2 x-1)=2(x-3) $$

Step-by-Step Solution

Answer

The solution is x = 3.

Step 1: Distribute the constants

First, distribute the -3 through the parentheses on the left side of the equation. This means you multiply -3 by each term inside the parentheses: -3(2x - 1) results in -6x + 3. For the right side, distribute the 2 through (x - 3): 2(x - 3) results in 2x - 6. Now the equation looks like this: 5x - 6x + 3 = 2x - 6.

Step 2: Simplify and combine like terms

Combine the like terms on the left side: 5x and -6x. This simplifies to -x. Now the equation is -x + 3 = 2x - 6.

Step 3: Move variables to one side

In order to solve for x, we need all the x terms on one side of the equation. Add x to both sides of the equation to get rid of -x on the left: -x + x + 3 = 2x + x - 6. This simplifies to 3 = 3x - 6.

Step 4: Move constant to other side

To isolate the x term, add 6 to both sides of the equation: 3 + 6 = 3x - 6 + 6. This results in 9 = 3x.

Step 5: Solve for x

To find x, divide both sides of the equation by 3 to isolate x: (9 / 3) = (3x / 3). This simplifies to x = 3.

Key Concepts

Linear Equations

Linear equations are equations of the first degree, meaning they involve only the variable raised to the power of one. In their simplest form, they have the structure \(ax + b = 0\), where \(a\) and \(b\) are constants, and \(x\) is the unknown variable.
Linear equations graph as straight lines on a coordinate plane, providing a straightforward way to see relationships between variables. They form the foundation for more complex algebraic equations.
To solve linear equations, you typically follow steps to isolate the variable on one side. This includes using properties of equality and algebraic manipulation such as adding, subtracting, multiplying, or dividing both sides of the equation by the same number. Linear equations are a key element in anlayzing and understanding relationships in algebra.

Solving Equations

Solving equations involves finding the value of the unknown variable that makes the equation true.
The goal is to find the value of the variable by isolating it on one side of the equation.

• This often requires moving terms from one side of the equation to the other.
• Performing arithmetic operations such as addition, subtraction, multiplication, or division.
• Using the inverse operations to simplify and solve the equation.

In our example, we had to perform several steps to solve the equation. We started by distributing numbers through parentheses. Then we combined like terms and moved variable terms to one side to simplify the equation. The final steps involved moving the constant to one side and isolating \(x\) by dividing both sides by the coefficient of \(x\).

Distributive Property

The distributive property is a fundamental algebraic principle used to simplify equations. It involves distributing or multiplying a single term across terms inside a set of parentheses. Mathematically, it is expressed as \(a(b + c) = ab + ac\).
In our equation, this property was applied twice:

• On the left side: \(-3(2x - 1)\) was expanded to \(-6x + 3\).
• On the right side: \(2(x - 3)\) was expanded to \(2x - 6\).

By using the distributive property, we can view and manage each part of the equation more clearly, aiding in combining like terms and simplifying the equation.

Combining Like Terms

Combining like terms is a crucial skill in algebra for simplifying expressions and solving equations. Like terms are terms in an equation that have the same variable raised to the same power. Only these terms can be added or subtracted.
In the equation given, the terms \(5x\) and \(-6x\) were combined to simplify the expression to \(-x\) on the left side of the equation.

• This simplification helps in organizing the equation.
• Makes it easier to isolate the variable during solving.

By reducing the equation to fewer terms, combining like terms reduces complexity, making the final isolated variable easier to solve for. This technique is used frequently when dealing with linear equations to streamline the problem-solving process.