I think the easiest way to solve these equations with fractions is to multiply both sides by a common denominator. The denominators are 5, 15, and 3, so if I multiply by 15, I'll eliminate the fractions. Most people don't enjoy working with the fractions, so eliminating them sounds like a great idea.
We can do this because we've got an equal sign. This is an equation. So if I multiply both sides by 15, it's not changing this solution. I'm going to multiply each.
term by 15, so I'm distributing that 15. I'm going to take it a piece at a time, just this chunk. So I'm going to think of 15 as over 1. Okay, for that first piece, 5 will go into 5 once, 5 will go into 15 three times. Now I'm going to do 3 times x minus 8. I just write that down, because I need to distribute it. Okay, take the next chunk. 15 will go into 15 once, 15 will go into 15 once, I'm left with just 8. And on this side, 3 will go into 15 5 times, and I get negative 5x.
So I need to distribute my 3, 3x minus 24 plus 8 equal negative 5x. Your next step is to combine like terms on each side of the equal sign. That is negative 16. Now I want to get my variables together, so I'll subtract 3x from both sides to move that 3x over. 3x minus 3x is 0. That negative 16 equals negative 8x. I divide by negative 8. We get x is 2. Okay, same idea on number two.
We've got fractions on both sides. If I multiply by the common denominator of 10, all the fractions will be gone. This one on the right side, you could combine those.
I think it's easier just to go ahead and get rid of the fractions first. Maybe less confusing. Okay, we're going to put all our 10s over 1, and we're going to take this piece.
5 will go into 5 once, 5 will go into 10 twice. We've got 2 times 2x plus 5x chunk. 2 will go into 2 once, 2 will go into 10 five times. We've got 5 times 3x plus 1. Last piece.
2 over 1 to 2, 1's 2 over 1 to 10, 5 times. We've got 5 times negative x plus 7. Okay, distribute your 2. 4x plus 10 equal, distribute the 5x plus 5. We've got 5 times negative x, so minus 5x plus 35. Next you want to combine like terms. There's nothing to do on this side.
15x minus 5x is 10x. 5 and 35 is 40. I'm going to subtract 10x from both sides. Write variable all on the left side.
I'll subtract 10 from both sides. We have negative 6x equals 30. Divide by negative 6. We get x is negative 5. Number 3 has a whole number on the right side. You're still going to multiply everything by a common denominator.
So the denominator of a whole number is 1. So I only have the denominators 3 and 4. So the common denominator is 12. Multiply 12 times x over 3. 12 times 1 fourth. But don't forget, you have to also multiply that side. Even though it doesn't have a fraction, you've got to multiply both sides.
Or you're changing the left side and not changing the right side. Okay, so we'll put all these 12s over 1, except that one. Take them one term at a time. 3 over 1 to 3 once, 3 over 1 to 12, 4 times, 4 times x is 4x. 4 over 1 to 4 once, 4 over 1 to 12, 3 times, 3 times 1 is 3. 12 times 7 is 84. Okay, there's no like terms to combine, so I'm going to solve for x by moving the 3 over.
84 minus 3 is 81. Write both sides by 4. x is 81 fourths.