Now is where we really start diving into some more complex topics. In this lesson, you will be learning to solve two-step equations, where you will have to utilize inverse operations as seen in lesson 1.3, but two times rather than just once (as the name obviously suggests. This lesson ramps up the difficulty a good amount more than the last one, so there will be many more examples to try and familiarize you with the topic, as well as a few more questions on the problem set. In this case, there is not much that needs to be explained in broad terms, but rather seen in action with examples, so let us get into those. One disclaimer is that PEMDAS does not really apply to this, and doing addition/subtraction first will make the process much simpler than multiplication/division.
1: 2x+5=24
In this equation, we must first utilize the inverse operation of addition (subtraction) to cancel out the 5 being added to 2x.
Once subtracting 5, we are left with the equation 2x=19, which is a simple one-step equation. We can apply the inverse operation of multiplication ( due to an assumed multiplication sign between 2 and x), which is division, then dividing 2x by 2 to get x and dividing 19 by 2 to get the final answer of 9.5.
We can confirm this by substituting 9.5 in place of x, giving us 2(9.5)+5=24, which is true.
Answer: 9.5
2: 8-4x=31
For this equation, a better way to think of it is as -4x+8=31, due to being able to utilize the commutative property of addition, with being able to also have the equation as 8+(-4x), allowing us to use the property.
We can first use the inverse operation of addition to then subtract 8 from both sides of the equation, leaving us with -4x=23.
Then, we can use division due to it being the inverse operation of multiplication (with the assumed multiplication between -4 and x) to then divide 23 by -4 to get the quotient of -5.75 as our final answer.
Answer: -5.75
3: 12x²=132
Firstly, lets do the inverse operation of multiplication which is division to get rid of the 12 and make the 132 a smaller (and therefore more manageable) number, simply giving us x²=11.
Secondly, we can simply find the square root of 11 (and cancelling out x² to simply x) to get our final answer of approximately 3.3.
Using the square root to cancel out the exponent works because they are practically the opposite of each other. For example, if I had 5², I could cancel it out by finding the square root of 25 (aka 5²), which is, of course, 5.
Answer: 3.3
4: -9+0.5x=92
Firstly, we are able to make this equation more straightforward visually, just by making it 0.5x-9 due to adding a negative, which is the same as subtracting. Now, we have the equation 0.5x-9=92.
Next, we can apply the inverse operation of subtraction (addition) to cancel out the x and further simplify the equation to 0.5x=101.
Finally, we can divide 101 by 0.5, which is the same as multiplying something by 2 (such as how we would get rid of a fraction like 1/2 by multiplying the reciprocal, which is 2/1) to get our final answer of 202.
Answer: 202
5: x²+31=44
Due to this one being set up in a nice way already, we can do our first step, which is, of course, using the inverse operation of subtraction to then cancel out the 31 and get the difference of 13 on the other side.
Secondly, the square root can be utilized as the inverse of operation of the exponent applied to x, then also square rooting 13, to get the final answer of 3.61.
Answer: 3.61
6: 31.5+√x=42
First of all, the commutative property of addition can be used, making the equation √x+31.5=42.
Next, the inverse operation of addition (subtraction) can be utilized to cancel out the 31.5 and simplify the equation to √x=10.5.
Finally, the inverse operation of square rooting (squaring) can be applied to then cancel out the square root symbol, and give the final answer of 110.25
Answer: 110.25
7: 3.14x÷4=82.2
Firstly, the inverse operation of division can be utilized (multiplication), therefore cancelling out the 4 and creating the product of 328.8, simplifying the equation to 3.14x=328.8.
Secondly, the inverse operation of multiplication (division) can be applied because of the assumed multiplication of x by 3.14, therefore giving us the final answer of 104.7.
Answer: 104.7
8: ˣ⁄₃+19=22
While either operation can be cancelled out first, I usually first cancel the one out that is "further" from x (which in most cases is addition or subtraction), though the order does not necessarily matter.
Firstly, the inverse operation of addition (subtraction) will be used to cancel out and create the difference of 3, simplifying the equation to ˣ⁄₃=3.
Secondly, the inverse operation of division (multiplication) can be applied, cancelling out the denominator and giving the final answer of 9.
Answer: 9
9: 21-4.3x=15
Firstly, let's rewrite this to -4.3x+21=15 by utilizing the commutative property of addition.
Next, 21 can be cancelled out by applying the inverse operation of addition (subtraction) to simplify the equation to -4.3x = -6.
Finally, the inverse operation of multiplication (division) can be used to give the final answer of 1.4.
Answer: 1.4
10: -92+x²=-30
Using the commutative property of addition, the equation can be swapped to x²-92=-30.
The inverse operation of subtraction (addition) can then be utilized to add 92 to both sides of the equation, simplifying it to x²=62.
Finally, the inverse operation of squaring (square rooting) can be utilized to give the answer of 7.87.
Answer: 7.87
11: 14-²⁄₇x=33
The commutative property of addition can first be applied to make the equation -²⁄₇x+14=33.
Next, the inverse operation of addition (subtraction) can be utilized to further simplify the equation to -²⁄₇x=19.
Finally, the inverse operation of multiplication (division) can be utilized, multiplying both sides by the reciprocal of -²⁄₇, which is -⁷⁄₂, giving the final product of -66.5.
Answer: -66.5
12: x³-22.1=19.77
Firstly, the inverse operation of subtraction (addition) can be applied to cancel out the 22.1 being subtracted with x³ as well as being added to 19.77, simplifying the equation to x³=41.87.
Since the expression is being cubed, the normal radical sign can not be utilized as the inverse operation, but rather the cube root symbol, which looks like ∛
Using the cube root, the final answer can be found, which is 3.47. Additionally, if the cube root is being applied to the variable, cubing it would be the inverse operation utilized.
Answer: 3.47
1: 18-2x=29
2: ˣ⁄₄·2=40
3: 49-3.3x=37
4: 9x÷12=6.22
5: 1x-9=0
6: ∛x÷4=2
7: 33x+44=55
8: 9x²=36
9: x³-90=11
10: √x+6.28=9
11: ˣ⁄₉+45=99
12: 11x÷12=0
13: ⁷¹⁄₁x+0.22=94
14: ˣ⁄₇-9=11
15: -7.34x+-1=0.5
1: x=-5.5
2: x=80
3: x=3.63
4: x=8.29
5: x=9
6: x=512
7: x=0.33
8: x=2 (or -2)
9: x=4.66
10: x=7.4
11: x=486
12: x=0
13: x=1.321
14: x=140
15: x=-0.07