Problem 56 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve equation and check your proposed solution in. \(0.02(x-2)=0.06-0.01(x+1)\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = 0.33\), and this solution has been verified as correct by back substitution into the original equation.

Step 1: Simplification

Simplify the equation to isolate the terms with the variable on one side by expanding the product on both sides: \[0.02x-0.04 = 0.06-0.01x-0.01\] Then, rearrange to collect like terms: \[0.03x = 0.01\]

Step 2: Solving

Next step is to solve the equation for 'x'. To isolate 'x', divide both sides by 0.03: \[x = \frac{0.01}{0.03}\] After performing the division, we get \(x = 1/3 = 0.33\) (rounded to 2 decimal places)

Step 3: Checking

Finally, substitue the value obtained for x in the original equation: \[0.02(0.33 - 2) = 0.06 - 0.01(0.33 + 1)\] After performing the operations, we get (rounding to two decimal places) -0.03 = -0.03, which is true. Therefore, the solution \(x = 0.33\) is verified to be correct, since it satisfies the original equation

Key Concepts

Algebraic Simplification

Algebraic simplification is a process that makes complex algebraic expressions easier to work with. This involves combining like terms, using the distributive property to remove parentheses, and reducing fractions to their simplest form if necessary.

When faced with an equation like 0.02(x-2)=0.06-0.01(x+1), algebraic simplification would first address the multiplication within the parentheses. This means applying the distributive property: multiply 0.02 by both x and -2, and -0.01 by both x and 1.

As a result, the equation simplifies to 0.02x - 0.04 = 0.06 - 0.01x - 0.01. The next step is to combine like terms, which in this case means moving all terms containing x to one side and the constant terms to the opposite side, leading to 0.03x = 0.01. Simplifying our expressions in this way lays a clear path toward solving for the variable.

Isolation of Variables

Isolating the variable is a critical step in solving linear equations. It involves manipulating the equation so that the variable you're solving for is by itself on one side of the equation. This isolation allows us to find the value of the variable.

In the provided equation, once simplified to 0.03x = 0.01, the next logical step is to isolate x. To do this, you'd divide both sides of the equation by the coefficient of x, which is 0.03. By performing this operation, x is left alone on one side: x = 0.01 / 0.03 or x = 1/3.

This step is where the 'isolation' part truly happens. It is essential for students to become comfortable with this concept, as it applies to all forms of linear equations, whether simple or complex.

Solution Verification

Verification is the final, but an extremely crucial step in solving equations. It ensures that the solution is correct by substituting it back into the original equation to see if it holds true.

For instance, with the solution x = 0.33 for our equation, we perform a check by replacing x with 0.33 in the original equation: 0.02(0.33 - 2) = 0.06 - 0.01(0.33 + 1). Simplifying both sides should give us an identity, and in this case, it does, with both sides equal to -0.03.

Students should always verify their answers, as it not only confirms the correctness of their solution but also reinforces their understanding of the equation's behavior. It acts as a failsafe mechanism to catch any arithmetic or simplification errors made during the solving process.