Study Guide

Equations and Inequalities - Solutions to Equations

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Solutions to Equations

What is a Solution to an Equation?

Let's start with the basics: An equation is a string of mathematical symbols stating the equality of two expressions.

Sample Problems

1. The equation x = 5 means "x equals 5."

2. "Two times x plus seven equals three times x" can be written in symbols as 2x + 7 = 3x.

3. The equation   states that the quantities  x/2 – 2 and 10 are equal.

An equation is a claim that two quantities are equal. It's like if someone's all, "Hey man, flammable is the same as inflammable," and you're like, "no way, prove it." Think of that equal sign as "is," so when x = y – 5, we're saying that x is y – 5. An equation is true if those two quantities actually are equal, and it's not true ("false") if those two quantities are unequal. In our example of the gentleman who seems to be concerned about a fire hazard, we would write "flammable = inflammable." If these two terms are exactly equal, then the equation is true.

As it turns out, they are the exact same thing. Probably good to know.

Sample Problems

1. The equation 3 + 1 = 2 + 2 is true, since 3 + 1 and 2 + 2 are both equal to 4.

2. The equation 1 = 0 is false because 1 and 0 aren't the same value.

When an equation contains a variable, the equation may be true for some values of that variable and false for others. A solution to an equation is a value that, when substituted for the variable, makes the equation true. We could say "a piece of fruit drawn at random out of this box is a banana." If some fruits in the box are bananas and some are not, then this statement may sometimes be true and sometimes not. Hopefully true in this case, since potassium is an important part of a balanced diet.

Sample Problems

In the equation 2x = x + 4, we can tell that x = 4 is a solution since 2(4) does equal (4) + 4.

On the other hand, x = 3 is not a solution to the equation 2x = x + 4, since 2(3) doesn't equal 3 + 4.

  • Checking Solutions to Equations

    How Do We Check if a Value is a Solution to an Equation?

    An equation works like this:

    (left-hand side expression) = (right-hand side expression)

    Therefore, an equation is only true if the left-hand side expression actually does equal the right-hand side expression. Take a look at your hands. Are they exactly the same?

    It's totally normal if they aren't. Mostly, anyway. Um, maybe we'll come back to this line of reasoning later.

    To check if a given value is a solution to an equation:

    1. Evaluate the left-hand side expression at the given value to get a number. 
    2. Evaluate the right-hand side expression at the given value to get a number.
    3. See if the numbers match.

    Hey, it's matching! You do this with your socks every day. Sometimes not well, but at least the process is a vaguely familiar one.

    If the numbers you get from evaluating the two expressions are the same, then the given value is a solution of the equation (makes the equation true). If the numbers don't match, the given value is not a solution of the equation (makes the equation false). Take those values that aren't solutions and dump them right in the trash, because we won't be needing them any longer. Actually, maybe rinse them out and put them in with the recycling instead. We're trying to be green.

    Sample Problem

    Is x = 5 a solution to the equation

    The Not So Awesome Way (aka The Wrong Way)

    If the first thing we do is write down  we're making a claim without having done the work to see if the claim is true. Oh snap.

    The statement that the left-and right-hand sides are equal should come after evaluating the left-hand side, evaluating the right-hand side, and comparing the answers. If we were lawyers, we'd call this our "due diligence." Luckily, this is Shmoop Algebra, and we're manfully resisting the urge to make horrible lawyer jokes right now.

    The Super Awesome Way (aka The Right Way):

    Evaluate the left-hand side for x = 5 first:

    Then evaluate the right-hand side for x = 5:

    Since 2 = 2, we can say that x = 5 is a solution to the equation. Bet knowing this will help you sleep better tonight.

  • Number of Solutions to an Equation

    How Many Solutions Can an Equation Have?

    Okay, algebristas,* different equations can have different numbers of solutions. Right now, we're only counting solutions from the real numbers. Some equations have solutions that are imaginary numbers, but we'll get to those later. No, you don't need to send us a Thank You card. Your words are enough.

    *You know, people who serve mathuccinos. Alternatively, you.

    Sample Problems

    • The equation x = 5 has only one solution: the number 5.
    • The equation z2 = 4 has two solutions: z = 2 and z = -2.
    • The equation x = x has infinitely many solutions: any value of x will work, since x is always equal to itself.
    • The equation y2 = -5 has no real number solutions because the square of any real number is positive.

    We interrupt this program to bring you a History Snack. Don't let it ruin your appetite.

    Diophantus was a famous mathematician who's sometimes called "The Father of Algebra." Although with the way he got around, who wasn't he the father of? #ancientgossip

    Anyway, D-man only liked positive rational solutions to equations. He would probably call a lot of the equations we'll be solving "absurd" since they have negative solutions. It's hard to blame him, though—after all, there's no such thing as a negative child. (He wishes. Oh, burn.)

  • Equivalent Equations

    Two equations are said to be equivalent if they have exactly the same solutions.

    Sample Problems

    • The equations x = 1 and 2x = 2 are equivalent, since the only solution to either equation is 1.
    • The equations 2x = 0 and 3x = 0 are equivalent, since the only solution to either equation is 0.
    • The equations x = 1 and x = 2 are not equivalent, since their solutions, 1 and 2, aren't the same.

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