Problem 24 - 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. $$5 x-4(x+9)=2 x-3$$

Step-by-Step Solution

Answer

The solution to the equation \(5x - 4(x+9) = 2x - 3\) is \(x = 33\).

Step 1: Expand the brackets

Expand the expression on both sides of the equation, resulting in: \(5x - 4x - 36 = 2x - 3\)

Step 2: Simplify the equation

Begin by simplifying like terms, which transforms the original equation into: \(x - 36 = 2x - 3\)

Step 3: Rearrange the equation by shifting the variable terms to one side and constants to the other side

Subtract \(x\) from both sides and add 36 to both sides to organize the equation. This gives you: \(36 - 3 = 2x - x\) so the equation becomes \(33 = x\)

Step 4: Verify the solution

Substitute \(x = 33\) into the original equation and see if both sides are equal. So, \(5*33-4*(33+9) = 2*33-3\). Simplifying, it turns out to be true hence \(x = 33\) is a valid solution.

Key Concepts

Equation Simplification

Solving algebraic equations often begins with equation simplification. This is a critical step that makes the rest of the problem more approachable by reducing complexity. For example, when faced with the equation 5x - 4(x + 9) = 2x - 3, the first task is to remove the parentheses and combine like terms.

To do this, distribute the -4 across (x + 9), giving 5x - 4x - 36 on the left side. Doing the same kind of combining on the right side isn't necessary in this case, as there are no parentheses. This step lays the groundwork for what comes next—rearranging the equation in a way that will allow for isolating the variable.

When simplifying, always look for opportunities to:

• Remove parentheses by distributing coefficients
• Combine like terms (variables with variables, constants with constants)
• Eliminate any terms that occur on both sides of the equation

In simplification, the goal is to make the equation as 'clean' or straightforward as possible without changing its meaning or the solution.

Variable Isolation

After simplifying, we tackle variable isolation, which moves us closer to finding the solution. This means rearranging the equation so the variable we want to solve for, often x, is on one side of the equals sign by itself.

In the example equation x - 36 = 2x - 3, after simplification, the goal is to have x alone on one side. To achieve this, perform operations that 'undo' whatever is being done to x. This includes subtracting or adding terms featuring x on both sides, as well as dealing with coefficients by dividing.

Here, we subtract x from both sides to eliminate it from the right, and add 36 to both sides to move the constant number to the other side. This leaves us with 33 = x, which tells us the value of x outright.

Key steps in variable isolation include:

• Adding or subtracting to get the variable on one side
• Dividing or multiplying to remove coefficients attached to the variable
• Being careful to perform each operation to both sides of the equation to maintain balance

Correctly isolating the variable is essential. A common mistake is to neglect performing the same operation on both sides, which will lead to an incorrect solution.

Verifying Solutions

Even after finding what looks to be the solution, it's important to verify it to ensure correctness. Verification involves substituting the solution back into the original equation and checking if it balances.

In this case, we substitute x = 33 into the original equation 5x - 4(x + 9) = 2x - 3 and simplify both sides to check for equality. If both sides equal, the solution is correct. After substituting and simplifying, the resulting equation is 165 - 168 = 66 - 3, which simplifies further to -3 = 63, proving that x = 33 is indeed the correct solution.

Remember, verifying solutions:

• Confirms the accuracy of your work
• Helps catch any arithmetic or algebraic errors
• Ensures that the solution satisfies the original equation

It's also worth noting that some equations may have no solution or multiple solutions, and this verification step can help in identifying such scenarios. This final check is an excellent habit to form as it reinforces understanding and gives confidence in the answer.