Problem 54 - Algebra 1: Concepts and Skills

Question

Which equations are equivalent? I. \(3 x-4 x+18=5 x\) II . \(4+6 x=8 x-2\) III. \(2 x-8=7-x\) (A) I and II \(\quad\) (B) II and III \(\quad\) C All \(\quad\) (D) None

Step-by-Step Solution

Answer

The equivalent equations are I and II.

Step 1: Simplify each equation

Start with the first equation \(3 x - 4 x +18 = 5 x\), combine like terms to get \(-x + 18 = 5x\). For the equation II: \(4+6 x=8 x-2\), rearrange to give \(6 x - 8 x = -2 - 4\). For the equation III: \(2 x - 8 = 7 - x\), rearrange to obtain \(2x + x = 7 + 8\).

Step 2: Solve for x in each equation

For the first equation, solve for \(x\) by adding \(x\) to both sides to get: \(18 = 6x\), then divide by 6 to get \(x = 3\). For the second equation, simplify the left-hand side to get \(-2x = -6\), divide by \(-2\) to get \(x = 3\). For the third equation, simplify to obtain \(3x = 15\), divide each side by 3, you get \(x = 5\).

Step 3: Check if the equations are equivalent

Equations I and II both give \(x = 3\) and thus are equivalent. Equation III gives \(x = 5\), so it's not equivalent to the other two.

Key Concepts

Solving Equations

Solving equations is like a puzzle where you find the value of the unknown variable, usually represented by \(x\). The ultimate goal is to make the variable stand alone on one side of the equation. You often do this by using various operations such as:

• Addition or subtraction: These operations help in getting rid of constants or coefficients on one side of the equation.
• Multiplication or division: Once the terms involving the variable are isolated, these can manipulate the coefficient of the variable to make it equal to one.

Breaking down the equation to its simplest form is crucial. It is achieved by reducing the equation to only have necessary terms and variables. This involves moving like terms together and balancing both sides of the equation. In this manner, you can determine the exact solution for the equation, finding out exactly what value \(x\) must take to satisfy it.

Algebra

Algebra is a major part of mathematics that provides formulas and rules to work with numbers and variables. It allows us to generalize arithmetic operations and solve equations systematically. Here’s why algebra is important:

• Using Symbols: Instead of using specific numbers all the time, algebra uses symbols (like \(x\), \(y\), etc.) to describe general situations. This allows you to create and solve equations.
• Forming Equations: By understanding how to form algebraic expressions, you can easily convert word problems or real-life scenarios into mathematical equations.
• Understanding Relationships: Through equations and inequalities, algebra helps in understanding the relationship between quantities.

Algebra is not just about finding the value of \(x\) in an equation. It's a way to model real-world problems and find solutions through structured mathematical expressions.

Mathematical Equivalence

Mathematical equivalence is all about determining if different equations represent the same relationship or not. In simpler terms, two equations are equivalent if they have the same solution. Here's how you can tell:

• Identical Solutions: If both equations result in the same value for their variables, they are equivalent. For example, if two equations both simplify to \(x = 3\), they are said to be equivalent.
• Transformation: Sometimes you can manipulate one equation using algebraic properties to transform it directly into the other.
• Checking Consistency: Consistency means the outcome of these equations remains the same despite changes in form or presentation.

In the exercise provided, equations I and II both yield \(x = 3\), proving they are equivalent. Equation III resulting in \(x = 5\) shows it's not equivalent to the others, highlighting how similar processes can have different outcomes.