Problem 1 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve 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 to the equation \(5x + 3x - 4x = 10 + 2\) is \(x = 3\).

Step 1: Simplify the left side of the equation

Combine like terms on the left side of the equation: \(5x + 3x - 4x = 4x\).

Step 2: Simplify the right side of the equation

The right side of the equation simplifies to \(10 + 2 = 12\). So, the simplified equation is \(4x = 12\).

Step 3: Solve the equation for \(x\)

Divide each side of the equation by 4 to solve for \(x\): \(x = 12 / 4 = 3\).

Step 4: Check the solution

Substitute \(x = 3\) back into the original equation: \(5 \cdot 3 + 3 \cdot 3 - 4 \cdot 3\). Check if this equals to \(10 + 2\). If yes, the solution is correct, otherwise a mistake was made and the solution is incorrect.

Key Concepts

Combining Like Terms

In algebra, combining like terms is crucial for simplifying equations. Like terms are the terms in an equation that have the same variable raised to the same power. Coefficients, the numbers multiplying the variables, can differ. Here's an effective way to combine like terms:

• Identify terms that have the same variable part.
• Add or subtract the coefficients of these like terms.
• Keep the variable and its exponent unchanged.

In the original exercise, we combined the like terms on the left side, the terms containing 'x'. We added the coefficients (5, 3, and -4) to get a single term, 4x. This step reduces the complexity of the equation and makes the next steps clearer.

Simplifying Equations

Simplifying equations makes them easier to solve. After combining like terms, the next step is often to simplify both sides of the equation. This includes combining constant terms (numbers without variables) and performing any addition or subtraction.

• Look for constant terms that can be combined.
• Perform arithmetic operations to simplify.
• Keep the equation balanced by doing the same operation on both sides.

For example, we simplified the right side of the exercise equation from 10 + 2 to 12, leaving us with an equation 4x = 12 that is much less intimidating and closer to finding the value of x.

Checking Solutions Algebraically

After finding a potential solution to an equation, it's important to confirm it's correct by checking the solution algebraically. This involves substituting the solution back into the original equation to see if it creates a true statement.

Steps for Checking the Solution:

• Substitute the solution into the original equation.
• Simplify both sides of the equation as you did previously.
• Compare the two sides. If they're equal, the solution is correct!

Using the above example, after calculating that x equals 3, we would substitute 3 for x in the original equation and check if both sides are equal. If the operations lead to the same value on both sides of the equation, our solution, in this case, x equals 3, is verified.