Problem 20 - 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. \(3(3 x-1)=4(3+3 x)\)

Step-by-Step Solution

Answer

-5

Step 1: Redistribute and Simplify

The equation is \(3(3x-1)=4(3+3x)\). Start by redistributing the terms by multiplying 3 and 4 with their corresponding expressions. This will result in the equation \(9x - 3 =12 +12x\)

Step 2: Rearrange the terms

Move all the terms involving x to one side and the constant terms to the other side. This gives us: \(9x - 12x = 12 + 3\). Simplifying further, we get \(-3x = 15\)

Step 3: Isolate the variable

Finally, we isolate the variable x by dividing every term in the equation by -3, which gives us \(x =-5\)

Step 4: Verify the solution

Insert the value of variable x back into the original equation to verify the solution. This will result in: \(3(3*(-5) - 1) = 4(3 + 3*(-5))\). Simplifying both sides gives us \(-45 = -45\), confirming that the solution is valid.

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that express the equality between two algebraic expressions. They consist of variables, constants, and algebraic operations (such as addition, subtraction, multiplication, and division). A simple example is the equation 3x + 2 = 11, where 3x + 2 is an algebraic expression equal to the constant 11.

In solving algebraic equations, the goal is to find the value of the unknown variable that makes the equation true. The process often involves several steps, including simplifying expressions, distributing, combining like terms, and isolating the variable. A crucial skill in working with these equations is understanding the properties of equality and operations, which allow you to maintain balance in the equation as you manipulate it to find the solution.

Substitution Method

The substitution method is one of the techniques used to solve systems of algebraic equations. It involves replacing a variable with an equivalent expression from another equation in the system. This method is particularly useful when one of the equations can easily be solved for one of the variables.

To apply the substitution method in solving a single equation, you might first express the variable in terms of other terms in the equation, and then substitute back into the original equation to verify the solution. In the example provided, the solution process dealt with a single equation in one variable. However, in more complex scenarios with multiple equations and variables, the substitution could involve solving one equation for a variable and then substituting this expression into another equation to find the values of the remaining variables.

Equation Verification

After solving an algebraic equation, it is essential to verify the solution to ensure its correctness. Equation verification involves substituting the solution back into the original equation to check if the equality holds true. If both sides of the equation result in the same value, the solution is confirmed to be correct.

In the provided exercise, after finding that x = -5, the verification process made use of substitution: 3(3(-5) - 1) on the left side and 4(3 + 3(-5)) on the right side of the original equation. Simplification then showed that both expressions equaled -45, which assured that the solution x = -5 is indeed valid. This step is critical in problem-solving as it prevents the acceptance of incorrect solutions that may result from simple calculation errors.