Question

Tell whether the equations are equivalent. $$21 x=7 \text { and } x=3$$

Short answer

No, the equations are not equivalent because their solution sets are not the same. Their solutions are \(x = \frac{1}{3}\) and \(x = 3\), respectively.

Step-by-step solution

Solve the first equation

The first equation is \(21x = 7\). To solve for x, simply divide both sides by 21. This gives \(x = \frac{7}{21}\), which simplifies to \(x = \frac{1}{3}\)

Solve the second equation

In the second equation, \(x = 3\), x is explicitly given, and there are no further steps necessary to find its value.

Compare the solutions

Now, it becomes apparent that the first equation gives us the solution \(x = \frac{1}{3}\) whereas the second equation gives us \(x = 3\). As these values are not the same, the two equations are not equivalent.

Key concepts

Solving Linear Equations

Linear equations are the starting point for much of algebra and are as straightforward as they come. Essentially, a linear equation is one in which, if you were to graph it, you'd see a straight line. Now, the goal in solving a linear equation is to find the value of the variable that makes the equation true.

Take, for instance, the equation we are looking at: \(21x = 7\). It’s a simple one-step equation where you need to perform the inverse operation to isolate the variable. In this case, we divide both sides by 21, because division is the inverse of multiplication and this helps us 'undo' the multiplication of \(x\) by 21. This leaves us with \(x = \frac{1}{3}\).

Understanding the basics of solving linear equations is vital because it lays the foundation for all other algebraic manipulations and problem-solving you'll encounter later on.

Simplifying Fractions

Simplifying fractions is an important skill because it often makes the numbers easier to work with. Think about it: would you rather have a pocketful of quarters or a large bag of pennies? Simplification can make math much cleaner. The process comes down to reducing the numerator and denominator to their smallest numbers while keeping the value of the fraction the same.

With the fraction \(\frac{7}{21}\), we notice that both 7 and 21 have a common factor, which is 7. By dividing the top and bottom by this common factor, we achieve the simplified fraction \(\frac{1}{3}\). It's like sharing a pizza evenly: if you cut it into 21 pieces and take 7, that's the same as if you cut it into 3 larger pieces and took just one. They’re equivalent portions – and with math, we always aim for the most straightforward expression possible.

Comparing Solutions

When comparing solutions, we're looking for a match, a pair of equivalent answers that signal our equations are effectively the same. It's a bit like trying to find two socks that make a pair in a drawer full of mismatched socks. With equations, we check to see if the solutions we derive make the two expressions interchangeable.

In our exercise, we find that one solution gives us \(x = \frac{1}{3}\) and another gives us \(x = 3\). These two values are as different as a quarter is to a dollar bill. Since the two solutions aren't the same, we can say with certainty that the equations are not equivalent. Recognizing when equations are equivalent is necessary for verifying solutions in more complex scenarios in algebra, such as systems of equations. Understanding the trick lies in accurate calculation and logical comparison.