 Hello, this is a video about linear equations, how to solve multi-step equations. To solve linear equations, there are a few steps you should always follow. The first step will always be to distribute and remove the parentheses. The second step is we'll be clearing fractions. The third step is that we will combine like terms on each side of the equation. We'll get the variable terms on one side, constant terms on the other. We'll simplify and solve, and if you want to, you can check your solutions, and this will tell you immediately if you got the answer correct or not. So our first multi-step equation we're going to solve has fractions in it, so eventually we will be removing those fractions. First thing I want to start off with though is to distribute. Multiply everything or distribute the one-half to everything in a first set of parentheses, and then distribute the one-third into everything in the second set of parentheses. 1 over 2 times 3x is going to give you 3x over 2. One-half times negative 1 will be negative one-half. One-third times x will give you one-third x, or you could just write x over 3. One-third times 2 will give you plus two-thirds. Now the next step is we will get rid of the fractions. So since I have two in the bottom of the fractions on the left side of the equal sign, three in the bottom of the fractions on the right side of the equal sign, I need to multiply by a number that is divisible by both two and three. The best option here would be six. So you can multiply by six or you could write six over one. Six times 3x does give you 18x. One times two does give you two on the bottom. Six times one gives you your minus six. One times two gives you two. Six times x is 6x over three. Six times two is 12. One times three is three. Now you can go through and clean this up a little bit and you'll get 9x minus three equals 2x plus four. This is definitely much more appealing to the I to solve than what we originally had. So there are no like terms on the left side of the equation. There are no like terms on the right side of the equation. So now we need to try to get our variable term on one side and everything else on the other. So in this specific example, I do have x's on both the left and right. So to move all the x's to the left hand side, I will take away 2x from both sides. This will give me 7x minus three equals four. Now we do need to get the variable term by itself. Since you have a minus three, the opposite of that would be two plus three on both sides, which you do to one side, you better do it to the other. 7x equals seven. Now to get x by itself, we are multiplying by seven. So we should divide both sides by seven and we will get x is equal to one. So the solution in this equation is one. Next equation, we will start off by distributing. Three times x is 3x, three times negative two would be negative six, and that is equal to 4x plus seven. We do have variables on both sides. There are no like terms on the left, no like terms on the right. So now the next step is to get the variables on one side, everything else on the other. So you can either take away 4x from both sides or take away 3x from both sides. I'm going to take away 3x from both sides. You get negative six equals x plus seven. You do want to get x by itself. We are adding seven to it, so you should take away seven from both sides. You'll get negative 13 equals x, which is the same thing as saying x equals negative 13. That's because it's something called the symmetric property of equality. Next example, it is a fraction on the left-hand side, so you can choose to multiply everything by x plus one, but I have an easier strategy we should use. I'm going to write seven and seven over one, and take note that I have now created something called a proportion. A proportion is whenever you have a single fraction by itself on the left, a single fraction by itself on the right. To solve a proportion, you can do something called cross multiply. Multiply the diagonals together and set them equal to each other. So I multiply seven x plus one together, that's seven times x plus one equals three times one. So this is going to give you seven x plus seven equals three. You do have one variable term and it is on the left-hand side, so you need to move the constant term to the other side by taking away seven from both sides. Seven x equals negative four. Last step is to divide both sides by seven. You'll get x equals negative four over seven. So next equation, start off by distributing. You'll get two x, two times four is plus eight is equal to two x plus seven. Now you have variable terms on both sides, so let's take away two x from both sides. You end up getting something such as eight equals seven, which unfortunately is something that is never true. When you get a statement that is never true, such as eight equals seven or any number is equal to some other number, the answer is no solution. So it doesn't matter how long you spent trying to solve this equation and trying to find what x is, there is no value of x that will satisfy this equation. On another note, if you obtain a solution or answer such as zero, zero, zero equals zero, something that is always true. It could be zero equals zero, seven equals seven, eight equals eight, a number is equal to itself. Anytime you obtain a solution like this, the answer is all real numbers. The answer to the equation would be all real numbers. So these are two very special cases for equations. So let's up the difficulty, yes, a little bit. So this is a rational equation that we still solve using the same strategies or approaches as we do for linear equations. And we can't really cross multiply because we don't have one fraction by itself on the right. Now you could take the two rational expressions on the right. Combine them into one fraction by finding a common denominator. Then you could cross multiply. But we're just going to multiply every single term by the least common denominator to get rid of the fractions, just like we multiplied by six earlier, which was divisible by both three and two. We are now going to multiply by x minus two, x minus one, because that would be the least common denominator of the three denominators present. Since x minus two is in the first denominator, it needs to be in your least common denominator. Since x minus one is in your second denominator, you need x minus one in your least common denominator. When you go to the last fraction, you notice that you have those x minus one and x minus two. This is a new fraction. It's the third fraction. Since those factors are already in the least common denominator, you do not need to throw more copies of them in there and complicate the situation. So we'll be multiplying by x minus two, x minus one. Before I move forward though, I want to make a note here that says notice that we have the variable in the denominator now. That's why it's a rational equation. So you have to be careful for values of x that cause the bottom of the fraction that equals zero. In this case, what causes x minus two to be zero when x is what number? That would be two. So x cannot equal two. And then looking at x minus one, x cannot equal one. These two values are not allowed for x. Okay, so let's take every numerator and distribute x minus two, x minus one to it. You get three times x plus x minus two times x minus one over the current denominator is equal to one times x minus two, x minus one over x minus one plus seven times x minus two, x minus one over the current denominator there. And it is at this step, it is at this step that you can now cancel out common factors in each fraction. First fraction, x minus two is cancel out leaving you a three times x minus one. Second fraction, x minus one's cancel you out leaving you as just x minus two. Fraction number three, both the x minus two and x minus one cancel out leaving you as plus seven. On the left side, we'll distribute to get three x minus three. And on the right side, we'll get x plus seven when we combine like terms. Let's get the x's all to one side. So let's take away x. This is like taking away one x by the way. So you'll get two x minus three equals five. Add three to both sides. We get two x equals eight. At the end of the day, after you divide both sides by two, you get a final answer of x equals four. Is four allowed as an answer to this equation? Yes, only one and two are prohibited. So x equals four is our final answer for this equation. So how about another one? All right. So in this situation, x cannot equal what causes x minus one to equal zero. That would be one itself. So we'll go ahead and get rid of the fractions by multiplying through by the least common denominator, which means multiplying through by just x minus one. All right. So fraction number x times x minus one over x minus one plus two times x minus one equals three times x minus one over x minus one. Now cancel out x minus ones cancel out in the first fraction. Next fraction or last fraction, the x minus ones do cancel out as well. So leaving you with three. Give it a multiply two with everything in the parentheses that follows. Two times x is two x. Two times negative one would be negative two. And that is going to equal three. We can combine like terms on the left side. We have a three x and a two x. That's five x minus two equals three. We want to get the x term by itself. So add two to both sides. Five x equals five. We have five x equals five. Since we're multiplying by five, we divide both sides by five and we actually get x equals one. But don't box that answer because guess what one causes in this fraction or in this equation? It causes the fraction, the bottom of the fractions to be zero. One minus one is zero. We already said x cannot equal one. So this is a bogus solution. Cross it out. There's actually no solution for our given equation here. There is no solution. So let's do one application question that uses linear equations and we'll be done. The perimeter of a rectangle is 100 yards. Draw me a rectangle here. Perimeter is 100 yards. Remember perimeter is distance around a polygon. What are the dimensions of the rectangle if the length is 30 yards more than the width? Well, that's a good question, right? Well, I know with rectangles, two sides are equal to each other and then the other two sides are equal to each other. One side is the length. The other side will be the width. So let's let the width equal x. So mark both sides of your rectangle with x. Then that would mean what? What would the length have to be if the width is x? We said that the length is 30 yards more than the width. So the length would have to be the width, which is x plus 30. So label the other two sides which are across from each other has x plus 30. You know that all four of these sides added together is equal to 100. If I set up that equation, solve for x, I will know the width. Then I can find the length. So I'm going to add up all four sides, x plus 30 plus x plus 30 plus x plus x is equal to 100. On the left side, we have a total of 4x plus 30 plus 30, which is 60. That's going to equal 100. We will take away 60 from both sides and we get 4x equals 40. Divide both sides by four and look what we get for x. A nice solid 10. So the width is 10. So it would be the length. It would be 10 plus 30. The length would be 40. So my dimensions of the rectangle would be 10 yards by 40 yards. So that is actually how you solve this application question. So I hope you enjoyed learning a little bit or reviewing a little bit about how to solve linear equations. Thank you for watching.