Comparison Method
Step 1: Isolate your “Y” in each equation (in order to obtain your two “Chunks”). This means that you want “Y” to be the only thing on the LEFT side of the equal sign – and remember, you only want one “Y”.
Ex: 1) 2x + 19y = 198 2) x – 23y = -226
19y = 198 – 2x -23y = -226 - x
19 19 -23 -23
(*remember you want to solve for “1y”)
Step 2: Now rewrite your “Chunks” side by side, separated by an equal sign (=). Cross-multiply your “Chunks” to solve for “X”.
Ex: 198 – 2x = -226 – x
19 -23
CROSS-MULTIPLY:
19 (-226 – x) = -23 (198 – 2x)
-4294 – 19x = -4554 + 46x
-19x – 46x = -4554 + 4294
-65x = -260
-65 -65
x = 4
Step 3: Now that you have solved for “X”, you will now solve for “Y” by plugging in the value found for “X” into ANY of the 2 original equations.
Ex: 2x + 19y = 198
2(4) + 19y = 198
8 + 19y = 198
19y = 198 – 8
19y = 190
19 19
y = 10
Elimination (By Addition) Method
Ex: 1) 2x + 19y = 198 2) x – 23y = -226
Step 1: What you want to do is make your “X” or “Y” values in both equations, equal.
In this example, if we multiplied the entire SECOND equation by 2, both of the equations “X” values will be equal, as desired.
Therefore the first equation will not change and the second will now be:
2x – 46y = -452
Step 2: Now that you have your “X’s” at equal value, in order to ELIMINATE THEM BY USING ADDITION, you will have to make one positive (+) and one negative (-). In this case both values of “X” are positive. Therefore I will choose EITHER equation and change all the signs.
I will use the second equation: (so now it will be…)
-2x + 46y = 452
Step 3: Now rewrite both equations, over/under each other, as such:
2x + 19y = 198
-2x + 46y = 452
Step 4: Add downwards (↓) (which will eliminate the “X’s”, as desired) and you will then solve for the value of “Y”. (one “Y”)
+ 2x + 19y = 198
↓ -2x + 46y = 452
65y = 650
(*now divide both sides by 65 to solve for ONE “Y”) 65 65
y = 10
Step 5: Now that you have solved for “Y”, you will now solve for “X” by plugging in the value found for “X” into ANY of the 2 original equations.
Ex: 2x + 19y = 198
2x + 19(10) = 198
2x + 190 = 198
2x = 198 – 190
2x = 8
2 2
x = 4
Substitution Method
Example: 1) 2x + 19y = 198 2) x – 23y = -226
Step 1: Isolate either the “x” or “y” in only one of the equations. (In this case, we will isolate the “x”.)
x – 23y = -226
x = -226 + 23y
Step 2: Substitute the expression you’ve obtained for this variable into the other equation – then it will become one equation with only one variable. (In this case, now solve for “y”.)
2x + 19y = 198
2(-226 + 23y) + 19y = 198
-452 + 46y + 19y = 198
46y + 19y = 198 + 452
65y = 650
65 65
y = 10
Step 3: Now choose either original equation and plug in the value we found for “y” and solve for “x”.
2x + 19y = 198
2x + 19(10) = 198
2x + 190 = 198
2x = 198-190
2x = 8
2 2
X=4
Solution: (4,10)
Step 1: Isolate your “Y” in each equation (in order to obtain your two “Chunks”). This means that you want “Y” to be the only thing on the LEFT side of the equal sign – and remember, you only want one “Y”.
Ex: 1) 2x + 19y = 198 2) x – 23y = -226
19y = 198 – 2x -23y = -226 - x
19 19 -23 -23
(*remember you want to solve for “1y”)
Step 2: Now rewrite your “Chunks” side by side, separated by an equal sign (=). Cross-multiply your “Chunks” to solve for “X”.
Ex: 198 – 2x = -226 – x
19 -23
CROSS-MULTIPLY:
19 (-226 – x) = -23 (198 – 2x)
-4294 – 19x = -4554 + 46x
-19x – 46x = -4554 + 4294
-65x = -260
-65 -65
x = 4
Step 3: Now that you have solved for “X”, you will now solve for “Y” by plugging in the value found for “X” into ANY of the 2 original equations.
Ex: 2x + 19y = 198
2(4) + 19y = 198
8 + 19y = 198
19y = 198 – 8
19y = 190
19 19
y = 10
Elimination (By Addition) Method
Ex: 1) 2x + 19y = 198 2) x – 23y = -226
Step 1: What you want to do is make your “X” or “Y” values in both equations, equal.
In this example, if we multiplied the entire SECOND equation by 2, both of the equations “X” values will be equal, as desired.
Therefore the first equation will not change and the second will now be:
2x – 46y = -452
Step 2: Now that you have your “X’s” at equal value, in order to ELIMINATE THEM BY USING ADDITION, you will have to make one positive (+) and one negative (-). In this case both values of “X” are positive. Therefore I will choose EITHER equation and change all the signs.
I will use the second equation: (so now it will be…)
-2x + 46y = 452
Step 3: Now rewrite both equations, over/under each other, as such:
2x + 19y = 198
-2x + 46y = 452
Step 4: Add downwards (↓) (which will eliminate the “X’s”, as desired) and you will then solve for the value of “Y”. (one “Y”)
+ 2x + 19y = 198
↓ -2x + 46y = 452
65y = 650
(*now divide both sides by 65 to solve for ONE “Y”) 65 65
y = 10
Step 5: Now that you have solved for “Y”, you will now solve for “X” by plugging in the value found for “X” into ANY of the 2 original equations.
Ex: 2x + 19y = 198
2x + 19(10) = 198
2x + 190 = 198
2x = 198 – 190
2x = 8
2 2
x = 4
Substitution Method
Example: 1) 2x + 19y = 198 2) x – 23y = -226
Step 1: Isolate either the “x” or “y” in only one of the equations. (In this case, we will isolate the “x”.)
x – 23y = -226
x = -226 + 23y
Step 2: Substitute the expression you’ve obtained for this variable into the other equation – then it will become one equation with only one variable. (In this case, now solve for “y”.)
2x + 19y = 198
2(-226 + 23y) + 19y = 198
-452 + 46y + 19y = 198
46y + 19y = 198 + 452
65y = 650
65 65
y = 10
Step 3: Now choose either original equation and plug in the value we found for “y” and solve for “x”.
2x + 19y = 198
2x + 19(10) = 198
2x + 190 = 198
2x = 198-190
2x = 8
2 2
X=4
Solution: (4,10)