In this topic, you will learn how to solve equations with variables on both sides. Yet again, this will be like adding on to the skills you already have learned with simplifying expressions and solving algebraic equations. In these equations, however, a variable like x, y, or z will now be on both sides, not one as you've seen before. However, these are very easy to deal with and only require one extra step. Firstly, you will still have to use the distributive property, combine like terms, and simplify the equation as needed, but now on both sides potentially. After that, you will then subtract the total number of x's from one side and then apply it to both sides of the equation. For example, if I had the equation 4x+3=5x+2, you could subtract 4x from both sides of the equation, then making it 3=x+2. That is the most simple equation you will see, and therefore the easiest way it can be explained. In this lesson, there will be a few examples as well as questions on problem sets that take longer to solve, not necessarily due to them having confusing/complex concepts, but rather because of the amount of simplifying that needs to be done. Now, let's expand on that idea by getting into some more complex examples!
1: 12x+9=3x+90
In this equation, we can see that while the right side has 3x, the left side has 12x. While either x can be subtracted, it will usually be easier to work with if the coefficient of x is positive, so we would subtract 3x in this case, making the equation 9x+9=90.
Now, it can be solved as a normal equation, using the inverse of addition and subtracting 9 from both sides to get 9x=81, and finally applying the inverse operation of multiplication, dividing both sides of the equation by 9 to get the answer of x=9.
Answer: x=9
2: -20-3.5x=8.5x+30
In this equation, we have -3.5x and 8.5x, so it would most likely be easier to cancel out -3.5x. In that case, 3.5x would be added to both sides due to having to do the inverse operation of subtraction since it's -3.5x, which then makes the equation -20=12x+30.
Next, the inverse operation of addition will be utilized to then isolate 12x and make the equation -50=12x. Finally, the inverse operation of multiplication will be applied, dividing both sides by 12 and therefore giving us x=-4.17.
Answer: x=-4.17
3: 4x(9-3)-14x+5=15x-11
In this equation, we will first have to simplify the left side. In the parentheses, 9-3 can easily be simplified to 6, which will multiply with 4x to get 24x. Additionally, 24x and -14x will then be combined as they are like terms to 10x, and 5 will be added, therefore making the equation 10x+5=15x-11.
Since 10x has a smaller coefficient than 15x, cancelling it out would most likely be easier, which would be achieved by subtracting -10x from both sides, making the equation 5=5x-11. The inverse operation of subtraction will then be used, adding 11 to both sides to get 16=5x. Finally, the inverse operation of multiplication will be applied, dividing both sides by 5 to get the solution of x=3.2.
Answer: x=3.2
4: -5.75(x+8-3x)+68.25=3x(2-5)-2x+9
In this equation, both sides of the equation will have to be simplified, so we can break them up into separate expressions and then set them equal to each other.
Firstly, we have -5.75(x+8-3x)+68.25. In the parentheses, x and -3x can be combined, therefore simplifying the parentheses to make the equation -5.75(-2x+8)+68.25. -5.75 and -2x can then multiply to produce the product of 11.5x, and -5.75 and 8 multiplying to get -46. -46 and 68.25 can be combined, finally simplifying the left side to 11.5x+22.25.
Next, 3x(2-5)-2x+9 must be simplified. The parentheses can yet again be simplified, this time to -3, which will then multiply with 3x to get -9x. -9x and -2x then combine to get -11x, along with the 9 being added on top to get -11x+9.
Now we have the equation 11.5x+22.5=-11x+9. The inverse operation of subtraction can then be used, adding 11x to both sides to make the equation 22.5x+22.5=9. The inverse operation of addition will then be applied, subtracting 22.5 from both sides, making the equation 22.5x=-13.5. Finally, the inverse operation of multiplication be utilized, dividing both sides by 22.5 to get our final answer of x=-0.6.
Answer: x=-0.6
5: 2(7x-11x+⅐·49)+7(-0.5)=6.3(7-8x)+2(3.1-x)
Yet again, this equation will most likely be easier if broken up into separate parts for each side to be simplified, then put back into one equation.
Firstly, we must simplify 2(7x-11x+⅐·49)+7(-0.5). Furthermore, (7x-11x+⅐·49) should be simplified, with 7x and -11x combining to get -4, and 49 multiplying by ⅐ to get 7. Now, we have 2(-4x+7)+7(-0.5). Next, 2 will distribute to both terms in parentheses, multiplying with -4x to get -8x, and multiplying by7 to get 14. The other parentheses just have the term -0.5, so it will simply multiply by 7 to get -3.5. Now, we have -8x+14-3.5, and 14 and -3.5 can be combined to fully simplify the equation to -8x+10.5.
Now, we have the other side of the equation to simplify, 6.3(7-8x)+2(3.1-x). When looking inside of both pairs of parentheses, we can see there are only 2 terms in each which do not combine, so we can simply distribute the coefficient of each set. 6.3 multiplies with 7 to get the product of 44.1, and multiplies with -8x to get -50.4x. In the other set of parentheses, 2 multiplies with 3.1 to get 6.2, and multiplies with -x to get -2x, giving us 44.1-50.4x+6.2-2x which then simplifies to 50.3-52.8x .
We now have -8x+10.5=-52.8x+50.3. In this case, I will use the inverse operation of subtraction to add 52.8x to both sides, giving us 44.8x+10.5=50.2, then the inverse operation of addition will be used to then subtract 10.5 from both sides to get 44.8x=39.8. Finally, the inverse operation of multiplication will be used, dividing both sides by 44.8 to get the final answer of x=0.89.
Answer: x=0.89
1: 7x+31=12x-14
2: 82+48x=64x-126
3: 12x+874=-15+47x+49
4: 6.25-19x+3.5=8.5x-33
5: -4+3x-16.5+5x=9x-8.8+6.3-4x
6: 47.25x-65+4.5x=1.75x+890+3x
7: 3(5x-2x)-7-4x=12x+11-3.5x
8: 34+2x(7·6-48)-9=18x+9-7
9: 2(18x-49÷7)+8x-3=9x-8+3(41·0.5-3x)-4x
10: 6.5(2x+8.7·4-7.35x)-3.7=7x(7-4.25+6÷12)+16
11: 7(4x-9+3.8x)-0.2·80=18(8.6-9x+11x+9)-5
12: 0.25x(5.5+6·15÷4)+6.5=4(3.25x+4.5÷6)+1.75
13: -8.75(6x+4·5-4x+6)+4=4(9x-8.5x+7.7+5)-33
14: -3+14(0.6x+7.9x+10·0.25)-7x=-8(7x-9.95+2.45x)+3.5
15: 7(2x+16-5x)-9(63÷9)=5.5(12-5.8)+3x(6÷12·-4)+2x
1: x=9
2: x=13
3: x=24
4: x=1.55
5: x=6
6: x=20.32
7: x=-5.14
8: x=0.77
9: x=1.5
10: x=3.6
11: x=21.01
12: x=0.22
13: x=-12.37
14: x=0.27
15: x=0.88