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University of Michigan - Ann Arbor

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at solving linear equations. When we solve an equation, our goal is to find out what a specific variable equals. We do this by isolating the variable in an equation we look at for the properties of equality we want. Whatever is on the left side of the equal sign t equal. Whatever on the right side and throughout the process of solving an equation, whatever is on the left side must equal what's ever on the right. And so you have to find out what the value of that variable is so that both sides are equal to each other. So let's look at a few examples of an equation. Let's look at a one step equation, and for that equation, we're gonna look at a plus four and 39. Hundreds equals 76. So we want to find out what number I would have to be when added to 44 39. Hundreds Thio equal 76. Find out we wanna isolator a so we're going to move our 4.39 We're gonna do that by doing this opposite. It's added, So we're going to subtract, and we're going to subtract four and 39 hundreds on both sides of the equation on the left side is going to cancel out because a positive for 39 hundreds, minus 4 39 hundreds equals zero. So that means on the left side I'm left with just in a on the right side. I have 76 minus four and 39 hundreds. And so now when I subtract those, I get 71 61 hundreds. So now let's look at some multi step equations for multi step equations. You're gonna do a few things. The first thing you're going to do is you're going to start off by simplifying both shots. Both sides of the equation are two separate expressions, and you wanna make sure that both sides are simplified to us few terms as possible. Secondly, just like with any equation you're gonna start by adding, are subtracting toe isolate are variable, and then finally, you're going to multiply or you're going to divide to find in that final step to find out what are variable will equal. So let's look at the equation to Tom's two X plus three Monas three Tom's for X minus five equals 22 So we need to simplify the right side because it's only 22. It's already simplify. So let's concentrate on this left side. Let's do distributive property first. So two times two X is for X two times three. It's six. So we've gotten rid of the first parentheses. Let's get into our second one. Negative. Three times four x It's negative. 12 X negative. Three times negative. Five is positive. 15. I noticed we've got some, like terms still on the left side so we can combine positive for X and negative 12 x, and they're going to give me a negative eight X. Then you can combine negative six and 15 and get 21 now. The reason why we want to simplify both sides is so that we don't have to constantly go back and forth and add and subtract every single number. It actually saves us a lot of time in solving these equations if we can simplify. So now let's go ahead and solve, because now we have a one step equation. Well, a two step equation so we can simply start by subtracting 21 because we're solving for X. So we're going to subtract 21. So that leaves with this negative eight x or 21 they're going to cancel 20 to minus 21 is one. We'll divide by negative eight for that final step. When I divide a number by itself, it's going to also cancel out because it now becomes one. So that leaves me with X equals negative 1/8. Let's look at another multi step equation. Let's look at negative 10 X plus three. Tom's four X modest tube equals six. So we want to begin by simplifying the left side of our equation. So a negative 10 X I'm gonna go ahead and just bring it down, and we're gonna do distributive property. Three times for X is 12 x and three times negative. Two is negative. Six equal six. So now that we have done distributive property, we want to combine any light terms. So we have a negative 10 x that we can combine with a positive 12 x negative 10 x and 12 x will give me two X minus six. Equal six notice. On the left side, I have one term that has available one term that doesn't so That means that the left side is now simplified. We can start by adding six to both sides. So two X equals six plus six is 12. Divide by two, so X will equal six. One more practice problem to do together we're gonna look at two Tom's two X minus one minus four. Tom's three X plus one equals two. We need to do distributive property on the left side. So two times two x is for X two times negative one is negative. Two negative four times three x is negative. 12 x and negative four times one is negative. Four. Now we can combine any like terms so we can combine for X and negative 12 x, and I'll get a negative eight x. We can also combine negative two and negative four and then we'll have negative six. We're gonna add six to the to You will cancel out on the left and leave me with negative eight X and to plus six is eight. We'll divide by negative eight and we can see now that X will equal negative one

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