Problem 20 - Algebra 1

Question

Tell whether the equations are equivalent. $$-4 x=44 \text { and } x=11$$

Step-by-Step Solution

Answer

The equations are not equivalent, as the solution for x in the first equation is -11, while in the second, it is 11.

Step 1: Solve the first equation

To solve the first equation -4x=44, you need to isolate x. This is accomplished by dividing both sides of the equation by -4. So, \(x = 44 / -4\).

Step 2: Calculate the value of x

By doing the division from Step 1, you obtain the result \(x = -11\).

Step 3: Compare the results

Now, compare the calculated value of \(x\) for the first equation with the given value of \(x\) for the second equation. The second equation directly states that \(x = 11\). As \(x = -11\) from the first equation does not equal \(x = 11\) from the second equation, you can conclude they are not equivalent.

Key Concepts

Solving Linear Equations

Understanding how to tackle a linear equation is like learning the secret recipe to a magical potion in algebra. A linear equation, at its core, is an algebraic statement where two expressions are set equal to each other, typically involving one or more variables, like the letter 'x'. For instance, in the equation \( -4x = 44 \), 'x' is what we want to uncover.

When solving linear equations, the main aim is to find the value of the unknown variable that makes the equation true. It requires a series of steps that can include adding, subtracting, multiplying, or dividing both sides of the equation equally. It's just like balancing a scale – whatever you do to one side, you must do to the other to keep it level. In our exercise, by dividing both sides by -4, we neatly unravel the equation to find that \(x = -11\).

This process equips us with the mathematical tools to confront more complicated algebraic expressions with confidence. Moreover, mastering linear equations opens doors to understanding more advanced realms of math and science.

Isolating Variables

When faced with an equation, imagine the variable you're solving for, let's say 'x', is the protagonist of a story. The objective is to isolate this character, to get 'x' all by itself on one side of the equal sign.

Isolating a variable is like clearing a stage for the main actor. You move everything else away until the spotlight is solely on 'x'. In the equation \( -4x = 44 \), 'x' is sharing the stage with -4. By dividing the entire equation by -4, we make the other numbers and symbols disappear into the wings, leaving \(x\) to take a solo bow as \(x = -11\).

It's like we told the story of 'x' from being surrounded by clutter to being the clear hero of the equation. This technique not just simplifies our math scenarios but also helps us to locate the variable without complications. Remember, isolating the variable is the key step in turning a perplexing puzzle into a simple solution.

Comparing Algebraic Expressions

Now, think of comparing algebraic expressions as judging a contest between two chefs. Each chef presents a dish, and the judge must decide: are these the same dish, just prepared differently, or are they different altogether?

In math, equivalent equations are like two versions of the same dish. They look different, but deep down, they represent the same numeric value. To figure out if two algebraic expressions are equivalent, we need to find the values of the variables and compare the results. If the values are identical, we've got a match! In our original exercise, the comparison was between \( -4x = 44 \) and \(x = 11\). After solving the first, we found out that \(x = -11\), which is definitely not the same as \(x = 11\). So, our judgment? These two are serving up completely different mathematical meals - the equations are not equivalent.

Understanding this concept is pivotal because it helps us recognize different forms of the same truth in mathematics, or to catch when two expressions are leading us to entirely different conclusions.