Let's look at our objectives for today. We will start by discussing what is an equation? We'll then look at the properties of equality that we used when solving equations. We'll look at the inverse operations used to solve an equation. And finally, we'll do some examples to see how to isolate the variable we want to solve for in the equation.
So let's look at what an equation actually is. An equation is a mathematical statement that two expressions, or quantities, have the same value. For example, 7 plus 8 equals 10 plus 5 is an equation. Both sides of the equation have a value of 15. The equal sign is used to state that the two quantities are equal, or have the same value. The left side is equal to 15 and the right side is equal to 15.
Equations are widely used throughout mathematics, sciences such as physics, statistics, and business. We also solving equations in daily life, such as when making purchases and determining income. Being able to solve simple and complex equations is a fundamental mathematical skill.
Let's look at our properties of equality. A variable, represented with a letter in an equation, is an unknown value we are trying to find. So there are several properties of equality that help us solve equations or determine the variable in an equation.
We first have the addition property, which says that if a equals b, and c is any number, then a plus c is equal to b plus c. Adding c on both sides of the equation still gives us a true statement.
The subtraction property says that if a equals b, and c is any number, then a minus c is still equal to b minus c. So subtracting c from both sides of the equation still gives us a true statement.
The multiplication property says that if a is equal to b, and c is any number, then a times c is equal to b times c. So here, multiplying by c on both sides of the equation still gives us a true statement.
And finally, the division property says that if a is equal to b, and c is any non-zero number-- we can't divide by 0-- then a divided by c is equal to b divided by c. So dividing by c on both sides of the equation still gives us a true statement.
So in general, these properties tell us that whatever is done on one side of the equal sign must be done on the other side in order to maintain an equation, or a true statement.
Let's look at our inverse operations. We use inverse operations to solve equations.
One pair of inverse operations are addition and subtraction, because they undo each other, or they cancel each other out. For example, if we start with 3 and we add 5, we get 8. However, if we start with 8 and subtract 5, we're back to our original value of 3. So adding 5 and subtracting 5 undo each other. We can also see this is true if we look at 12 minus 8, which will give us 4. But if we start with 4 and add 8, we're back to our original value of 12.
Multiplication and division are also inverse operations, because they also undo each other. For example, 2 multiplied by 7 will give us 14, but 14 divided by 7 will bring us back to our original value of 2. So multiplying by 7 and dividing by 7 undo each other. We can see this if we look at 18 divided by 6, which will give us 3. But if we start with 3 and multiply by 6, we're back to our original value of 18.
Squaring and taking the square root are also inverse operations, because they undo each other. For example, if we start with 7 and square it, we have 49. And if we take the square root of 49, we're back to our original value of 7. If we have 36 and we take the square root, we have 6. If we start with 6 and square it, we're back to our original value of 36.
So these are our inverse operations.
Finding the solution or solving most equations involves isolating a variable. To do this, we want to rearrange our equation so that the variable, or the unknown quantity, is by itself on one side of the equation, and everything else is on the other side. To rearrange our equation, we use the operations that are inverse of the operations appearing in the equation. Let's see what that looks like.
We want to solve the equation x plus 10 equals 25. We know we want to isolate the variable x on the left side of the equation. Because we're adding 10 to the variable x, we know we need to use the inverse operation to addition, which is subtraction, to cancel out the plus 10.
So we're going to subtract 10 on the left side of the equation, which means we need to subtract 10 on the right side of the equation. This will leave us with x on the left and 15 on the right. So our solution is x equals 15.
For our second example, we have 1/3 times x, which equals 4. Because the 1/3 is multiplying by x, we're going to use the inverse operation of division to cancel it out. So we divide by 1/3 on the left and on the right side of the equation. But this starts to look a little messy.
So another way to cancel out the 1/3 is to multiply by its reciprocal, which would be 3/1. Multiplying by 3/1 on the left side of the equation will give us 3/3 times x. And multiplying by 3/1 on the right side will give us 12/1. This will simplify to be 1 times x equals 12. Dividing by one on both sides of the equation will give us our final answer of x equals 12.
Let's look at our last example. Suppose Jaime scored 18 goals during 9 games of soccer. He wants to know how many goals he scored, on average, per game. So we can write an equation to represent the situation. We're going to multiply the 9 games of soccer by x, our variable, which here, is how many goals he scored per game. And we know that's going to be equal to a total number of 18 goals.
So to solve this equation, because the 9 is multiplying by the x, we're going to divide by 9 on both sides of our equation. This will simplify to be x on the left side of the equation, and 2 on the right side of the equation. So our solution is x equals 2, which means that Jaime scored, on average, 2 goals per game.
Let's go over our important points from today. Make sure you get these in your notes, so you can refer to them later.
An equation is a mathematical statement that two expressions, or quantities, have the same value. A variable, which is represented with a letter in an equation, is an unknown value we are trying to find or solve for. To solve for a variable, we want to isolate the variable on one side of the equation and move everything else to the other side. And we use inverse operations to isolate the variable that we want to solve for. Addition and subtraction are inverse operations, multiplication and division are inverse operations, and squaring and taking the square root are also inverse operations.
So I hope that these key points and examples helped you understand a little bit more about solving single step equations. Keep using your notes, and keep on practicing, and soon you'll be a pro. Thanks for watching.
A mathematical statement that two expressions or quantities have the same value.
An unknown value that we are trying to find.