TranscriptPractice Hello! Today we’re going to take a look at solving equations with variables on both sides. The solving process for these types of problems will look similar to more simple forms of solving equations, it will just require a few additional steps. So, let’s start by taking a look at an example. \(5x+9=3x-7\) To
solve problems like these, you will want to get all the variable terms on one side of the equation and all the constant terms on the other side of the equation. It doesn’t matter which side either one goes on, just make sure that all the variable terms are on one side and all the constants are on the other side. So, for this, I’m going to get the \(x\)-terms on the left side of the equation. So, to do that, we’ll subtract \(3x\) from both sides. \(5x-3x+9=3x-3x-7\) If you wanted to subtract \(5x\) from both sides and get the \(x\)-terms on the right side of the equation, that would be totally fine as well. \(2x+9=-7\) So now, to get the constants on the other side, we’re going to want to subtract 9 from both sides. \(2x+9-9=-7-9\) \(2x=-16\) And finally, to solve the equation, we need to get \(x\) by itself, so we do that by dividing both
sides by 2. \(\frac{2x}{2}=\frac{-16}{2}\) \(x=-8\) Let’s take a look at another problem. \(-2x+13=6x-31\) This time, we are going to solve for the variable on the right side. So, to do this, we’re going to start by adding \(2x\) to both sides of the equation — that will get our \(x\)-terms on the right side. \(-2x+2x+13=6x+2x-31\) \(13=8x-31\) Now, we want to get our constants on the left side so we’ll add 31 to both sides. \(13+31=8x-31+31\) \(44=8x\) And finally, to get \(x\) by itself, we divide both sides by 8. \(\frac{44}{8}=\frac{8x}{8}\) \(\frac{44}{8}=x\) Now, this fraction isn’t in most simplest form, so the best thing to do is to simplify it. The way that we can simplify this fraction is
by dividing both the numerator and denominator by 4.
Let’s take a look at one more problem before we go.
So, first we’re going to add \(9x\) to both sides. If you wanted to subtract 4x from both sides and solve for \(x\) on the right side of the equation, that would totally be acceptable as well. The reason I chose to add \(9x\) to both sides is because I don’t particularly like working with negative numbers when I don’t have to. So I’m adding \(9x\) so that I’ll have a positive number of \(x\)‘s on the left side. So, that’s why I chose that, but if you like negatives and want to do that, you are more than welcome to.
Now I’m going to subtract 17 from both sides of our equation.
So, notice we have a negative over here, so we’re dealing with negatives anyways, so it’s just up to you back at this first step if you want to have negative \(x\)-values or negative constants. Either way will get you the right answer. So to get \(x\) by itself and solve for \(x\) we’re going to divide by 13 on both sides.
I hope that this video gave you more confidence in solving equations. Thanks for watching and happy studying! Practice QuestionsQuestion #1: \(x=9\) \(x=15\) \(x=8\) \(x=17\) Show Answer Answer: To get the variable terms on the left side, subtract \(2x\) from both sides of the equation. \(4x+9-2x=25+2x-2x\) Now, subtract 9 from both sides. \(2x+9-9=25-9\) Finally, divide both sides by 2. \(\frac{2x}{2}=\frac{16}{2}\) \(x=8\) Hide Answer Question #2: \(x=-14\) \(x=12\) \(x=10\) \(x=-10\) Show Answer Answer: To get the variable terms on the left side, add \(7x\) to both sides of the equation. \(-5x-16+7x=-7x+4+7x\) Now, add 16 to both sides. \(2x-16+16=4+16\) Then, divide both sides by 2. \(\frac{2x}{2}=\frac{20}{2}\) \(x=10\) Hide Answer Question #3: \(x=-\frac{19}{2}\) \(x=\frac{19}{2}\) \(x=\frac{7}{2}\) \(x=-\frac{7}{2}\) Show Answer Answer: To get the variable terms on the left side, subtract \(2x\) from both sides of the equation. \(6x-11-2x=2x+27-2x\) Now, add 11 to both sides. \(4x-11+11=27+11\) Then, divide both sides by 4. \(\frac{4x}{4}=\frac{38}{4}\) \(x=\frac{38}{4}\) Reduce the fraction by 2, to write the answer in simplest form. \(x=\frac{38\div2}{4\div2}=\frac{19}{2}\) Hide Answer Question #4: 5 hours 3 hours 10 hours 8 hours Show Answer Answer: Let \(x\) be the number of hours you both work. You have a flat fee of $20 and are charging $10 per hour, the amount of money you make for working \(x\) number of hours is \(20+10x\). Your friend has a flat fee of $35 and charges $7 per hour, so the amount of money your friend makes for working \(x\) number of hours is \(35+7x\). To determine how many hours you each need to work to make the same amount of money, we need to set the two expressions equal to each other. \(20+10x=35+7x\) To get the variable terms on the left side, subtract \(7x\) from both sides of the equation. \(20+10x-7x=35+7x-7x\) Now, subtract 20 from both sides. \(20+3x-20=35-20\) Finally, divide both sides by 3. \(\frac{3x}{3}=\frac{15}{3}\) \(x=5\) Hide Answer Question #5: Show Answer Answer: Let \(x\) be the number days you each run after the week of running alone. Since you run 4 miles each day, you run \(4x\) miles in \(x\) days. Similarly, since your friend runs 3 miles each day, she runs \(3x\) miles in \(x\) days. During the week you each ran alone, you ran 16 miles, and your friend ran 23 miles. So, after the first week of running alone, you will have run \(16+4x\) miles, and your friend will have run \(23+3x\) miles. Now, set the expressions equal to each other to find how many days it will take for you and your friend to run the same number of miles. \(16+4x=23+3x\) To get the variable terms on the left side, subtract \(3x\) from both sides of the equation. \(16+4x-3x=23+3x-3x\) Now, subtract 16 from both sides. \(16+x-16=23-16\) Therefore, 7 days after you and your friend started running together, you both will have run the same number of miles. Hide Answer Return to Algebra I Videos 402497 How do you solve equations with Pronumerals on both sides?When a pronumeral is on both sides of an equation, remove the pronumeral term from the right-hand side of the equation by using inverse operations. Then continue to use inverse operations to solve the equation for the pronumeral.
How do you solve equation problems?A General Rule for Solving Equations. Simplify each side of the equation by removing parentheses and combining like terms.. Use addition or subtraction to isolate the variable term on one side of the equation.. Use multiplication or division to solve for the variable.. |