Problem 53 - Introductory Algebra for College Students

Question

Solve each equation using the addition property of equality. Be sure to check your proposed solutions. $$4 x+2=3(x-6)+8$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -12\)

Step 1: Distribute the Multiplication Inside the Parenthesis

Distribute the multiplication in 3(x-6) to each term inside the parenthesis. The equation becomes: \(4x+2=3x-18+8\)

Step 2: Simplify the Equation

Combine like terms on each side of the equation to simplify it. The equation simplifies to: \(4x+2=3x-10\)

Step 3: Isolate the Variable x

To isolate x, first subtract 3x from both sides to get \(x+2=-10\), then subtract 2 from both sides to get \(x=-12\)

Step 4: Check the Proposed Solution

Substitute x=-12 back into the original equation: \(4(-12)+2\)=3((-12)-6)+8. Simplify both sides to confirm that they are equal, confirming that x=-12 is indeed the correct solution.

Key Concepts

Solving Equations

Solving equations is all about finding the value of the unknown variable, usually represented as \(x\). It involves a series of steps to manipulate the equation, making \(x\) the subject of the formula. The key is to use the same operation on both sides of the equation to maintain equality. Here are some steps to consider:

• Identify the equation: Look at both sides and identify what needs to be done to isolate \(x\).
• Use inverse operations: Apply the opposite operation to cancel out terms and isolate the variable. For additions, use subtraction; for multiplications, use division, etc.
• Simplify at every step: Tackle the equation little by little, simplifying terms whenever possible.
• Check your solution: Substitute your solution back into the original equation to ensure it results in a true statement.

By following these steps, you ensure that the logic remains sound, leading to the correct solution.

Distributive Property

The distributive property is a fundamental algebraic tool used to eliminate parentheses and make equations simpler. It states that multiply a number by a sum is the same as multiplying each addend separately and adding the products. In mathematical terms:\[a(b + c) = ab + ac\]In our exercise, we had \(3(x-6)\). Applying the distributive property, we multiply 3 with \(x\) and 3 with \(-6\) resulting in:

• First multiplication: \(3 \times x = 3x\)
• Second multiplication: \(3 \times (-6) = -18\)

Thus, the expression expands to \(3x - 18\). This helps to eliminate the parentheses and simplifies the equation, making solving easier from this point forward.

Combining Like Terms

Combining like terms is a crucial step in simplifying equations. It involves merging terms that are similar, i.e., terms that have the same variable raised to the same power. In the process:

• Identify like terms: Look for terms on the same side of the equation that have the same variable.
• Combine: Simply add or subtract their coefficients as applicable. For instance, combining \(3x\) and \(x\) results in \(4x\).
• Consider constants: Constants, or terms without variables, should be combined to reduce clutter on equation sides.

In the solution, \(3x\) and \(4x\) are like terms, which we treated accordingly to simplify our equation further. The simpler the equation, the fewer steps required to solve it.

Substitution Method

The substitution method is a way to verify the solution you find when solving equations. After isolating the variable and determining its value, you can check whether the solution works by substituting the value back into the original equation. Here's how it works:

• Replace the variable: Substitute the solution back into the equation where the variable appears.
• Calculate both sides: Simplify both sides of the equation while keeping them balanced.
• Check for equality: If both sides of the equation are equal, your solution is correct. If they're not, recalibrate your calculations.

For example, using \(x = -12\) from our solution, we substitute \(-12\) into both sides of the original equation and ensure they equal after simplification. This confirms that our solution is accurate, thereby completing the problem.