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Solving Equations by Combining Like Terms

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[MUSIC PLAYING] Let's look at our objectives for today. We'll start with an introduction to solving equations with like terms. We'll then review how to combine like terms.

We'll review the properties of equations. We'll also review how to isolate a variable when solving equations. And finally, we'll do some examples solving equations with like terms.

Let's start by introducing when we combine like terms when solving equations. Solving both simple and complex equations is a skill that is fundamental to all levels of mathematics. Many equations will require simplification by combining like terms on either side of the equation before using inverse operations to solve the equation. Solving equations are used to solve problems in mathematics as well as science fields, such as physics or chemistry, when balancing chemical equations.

Now, let's review how to combine like terms. A "term" is a collection of numbers, variables, and powers combined through multiplication. A "coefficient" is the number in front of a variable that acts as a factor or multiplier. Only like terms can be combined with addition or subtraction. Like terms are terms with the same variable and variable power.

Here's an example. We have the expression 3x squared plus 3y squared plus 12x minus 2x. Here, the only like terms are 12x and negative 2x, so the subtraction sign in front of the 2x is the same as adding a negative 2x. These are the only two terms that can be combined with addition or subtraction because they are the only two like terms in the expression.

12x plus negative 2x can be simplified by adding their coefficients, which gives us 10x. Notice that the variable and exponent remain unchanged when we combine like terms. So our simplified expression is 3x squared plus 3y squared plus 10x.

Now, let's review the properties of equations. These properties are important to remember when solving equations. An "equation" is a mathematical statement that says that two expressions or quantities are equal or have the same value.

When we are solving equations, we use these several properties of equality. The addition property says that, if a equals b and c is any number, then a plus c is equal to b plus c. The subtraction property says that, if a equals b and c is any number, then a minus c equals b minus c.

The multiplication property says that, if a equals b and c is any number, then a times c is equal to b times c. And the division property says that, if a equals b and c is any non-zero number, then a divided by c is the same as b divided by c. Because of these properties of equality, what is done to one side of the equal sign must be done on the other side.

When solving an equation, we use inverse operations to undo operations in an equation and isolate the variable or unknown quantity we want to know. Our inverse operations are addition and subtraction, multiplication and division, and squaring and taking the square root.

So now, let's review and do an example of isolating a variable in an equation. We want to solve 3x minus 8 equals 2x plus 1. Solving an equation means isolating the variable on one side of the equals sign with everything else on the other side. So we use reverse order of operations to isolate the variable-- meaning we use the acronym PEMDAS backwards.

So addition and subtraction are undone before any multiplication or division is undone. So we start by adding 8 to the left side, which will undo the subtracting 8. And if we add it on the left side of the equal sign, we need to add 8 to the right side where it can be combined with the 1.

Now, our equation is 3x equals 2x plus 9. Next, we want to subtract 2x on the right side so that it cancels out the positive 2x. If we subtract 2x on the right side, we need to subtract 2x on the left side where it can be combined with a 3x. So now, our equation is 1x equals 9.

Finally, we want to divide by 1 on the left side to cancel out the 1 being multiplied by the x. This means we have to divide by 1 on the right side. This gives us our final answer of x equals 9. When solving an equation, the solution should be substituted back into the original equation to verify that it is, indeed, a solution.

Finally, let's do an example of solving an equation which requires combining like terms. We want to solve negative 3x plus 15 plus x equals 2x minus 35 plus 10. We start by combining our like terms on the left and on the right side of the equation.

On the left side, we can combine the like terms negative 3x plus x. Adding our coefficients, negative 3 and 1, gives us negative 2x. On the right side, we can combine the like terms negative 35 and 10. Negative 35 plus 10 equals negative 25. So now, our equation is negative 2x plus 15 equals 2x minus 25.

Now, we start to isolate our x variable by subtracting 15 on both sides of our equation. Subtracting 15 on the left side will cancel out the 15, and subtracting 15 on the right side can be combined with the negative 25. So now, our equation is negative 2x equals 2x minus 40.

We'll then move the 2x on the right side of the equation by subtracting 2x on both sides. This gives us negative 4x equals negative 40. We then divide by the negative 4 that is being multiplied by the x on the left side of the equation, and we need to divide by negative 4 on both sides, which gives us x equals 10.

We can make sure that we have found the correct answer by substituting our solution of 10 in for x into our original equation. Doing this gives us negative 3 times 10 plus 15 plus 10 equals 2 times 10 minus 35 plus 10. Simplifying this, starting with multiplication, gives us negative 30 plus 15 plus 10 equals 20 minus 35 plus 10. We can add and subtract from left to right, which would give us negative 30 plus 15 plus 10, which is negative 5 on the left side of our equation, and 20 minus 35 plus 10, which gives us negative 5 on the right side of our equation. This is a true statement, so our solution of x equals 10 is correct.

Let's look at 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 says two expressions or quantities are equal or have the same value. Because of the properties of equality, what is done on one side of the equation must be done on the other side.

When solving an equation, we use inverse operations to undo operations in an equation to isolate the variable or unknown quantity we want to know. Many equations will require simplification by combining like terms on either side of the equation before using inverse operations to solve. Only like terms can be combined with addition or subtraction. Like terms are terms with the same variable and variable power. And to combine like terms, we add or subtract their coefficients and leave the variable in power the same.

So I hope that these key points and examples helped you understand a little bit more about solving equations by combining like terms. Keep using your notes, and keep on practicing. And soon, you'll be a pro. Thanks for watching.

Formulas to Know

- Addition Property of Equality
- Division Property of Equality
- Multiplication Property of Equality
- Subtraction Property of Equality