Problem 1 - Introductory Algebra for College Students

Question

Solve each equation. Be sure to check your proposed solution by substituting it for the variable in the original equation. $$5 x+3 x-4 x=10+2$$

Step-by-Step Solution

Answer

The solution for the equation is \( x = 3 \).

Step 1: Combine Like Terms

Combine the x terms on the left side, and add 10 and 2 on the right side: \(5x + 3x - 4x = 10 + 2\). This gives \(4x = 12\).

Step 2: Solve for x

To isolate x, divide both sides by 4. This gives: \( x = 12 / 4\). Solving for \( x \) gives \( x = 3 \).

Step 3: Check the Solution

Insert \( x = 3 \) back into the original equation: \(5(3) + 3(3) - 4(3) = 10 + 2\). This simplifies to 15 + 9 - 12 = 12, and further simplifies to 12 = 12. The equation holds true, so \( x = 3 \) is the correct solution.

Key Concepts

Combine Like Terms

When solving linear equations, one of the first steps is often to combine like terms. This means to add or subtract terms that have the same variable raised to the same power. For instance, when we look at the equation 5x + 3x - 4x = 10 + 2, we see three terms that all have an 'x' without any exponents. They are called 'like terms' because the 'x' variable is the same in each.

To combine them, simply add or subtract their coefficients (the numbers in front of the variables): 5 + 3 - 4. This process simplifies the left side of the equation to 4x. Remember, you can only combine terms that are alike; terms with different variables or exponents are not combined in this step.

Isolate Variable

The term isolate the variable refers to getting the variable by itself on one side of an equation to find its value. In the given problem, after combining like terms, we have 4x = 12. To isolate x, we need to eliminate the coefficient of x, which is 4.

We do this by performing the inverse operation. Since x is multiplied by 4, we divide both sides of the equation by 4, resulting in x = 12 / 4 or x = 3. This operation has effectively isolated x, providing us with its value.

Check Solution Algebra

It is vital to check the solution of an algebraic equation to ensure that it is correct. This involves substituting the found value of the variable back into the original equation. In our problem, we determined that x = 3. To check it, we replace every x in the original equation with 3: 5(3) + 3(3) - 4(3) = 10 + 2.

When we carry out the operations on both sides, we should get an identical value if our solution is correct. After the calculation, we find that both sides equal 12; thus, the solution x = 3 holds true, confirming its correctness.

Algebraic Expressions

At the heart of solving equations lie algebraic expressions; these are combinations of numbers, variables, and operation symbols that represent a specific quantity. In our equation, 5x + 3x - 4x and 10 + 2 are both algebraic expressions.

Understanding how to work with these expressions is crucial for manipulating and solving equations. You should know how to properly combine like terms, isolate the variable, and perform various arithmetic operations within these expressions to solve for unknown variables.