Question 1088: 1. 4x+3y=1, x=1-y
2. 2x-y=6, -x+y=-1
3. 6x-y=3, 4x-2y=-2
4. 2x+3y=7, x=1-4y
5. 2x+3y=6, x-3y=-15
6. 7x-5y=4, y=3x-4
7. 2y-5x=-1, x=2y+5
8. 4x+3y=1, 3x+4y=10
9. 6x-5y=3, 4x+3y=21
10. x+y=4, 3x+4y=10
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Answer by AnlytcPhil(1808)   (Show Source):
You can put this solution on YOUR website! I'll just solve #2 and #9. All are similar to these two:
2. 2x-y=6
-x+y=-1
Substitution method:
Pick one of the unknowns and an equation that contains it. You can actually
choose either letter and either equation it's in. However things will be
simpler if one of the equations contains +x, -x, +y, or -y, and you choose
that letter and that equation. We have three good choices in this problem, so
I'll arbitrarily pick y and the second equation:
1. Solve the chosen equation for the chosen letter
-x + y = -1
y = -1 + x
2. Replace the chosen letter in the OTHER equation by the expression which
that letter equals from step 1. Solve for the OTHER letter.
2x - y = 6
2x - (-1 + x) = 6
2x + 1 - x = 6
x + 1 = 6
x = 6 - 1
x = 5
3. Substitute that value for that letter into the result of step 1:
y = -1 + x
y = -1 + (5)
y = 4
----------------------------------
9. 6x-5y=3, 4x+3y=21
1. Rearrange the terms of the equations so that like terms line up
vertically like this:
Ax + By = C
Dx + Ey = F
----------------------
6x - 5y = 3
4x + 3y = 21
----------------------
2. Get the LCM (least common multiple) of the coefficients of x in
absolute value.
The LCM of 6 and 4 is 12.
3. Multiply each equation by whatever number is necessary to
cause the coefficient of x to be equal to the LCD.
The equation 6x - 5y = 3 needs to be multiplied by 2 to make the
coefficient of x to be 12
2[ 6x - 5y = 3]
12x - 10y = 6
The equation 4x + 3y = 21 needs to be multiplied by 3 to make the
coefficient of x to be 12
3[4x + 3y = 21]
12x + 9y = 63
Since these both have +12x I will pick the second one to multiply
through by -1, getting
-12x - 9y = -63, so now we have this system:
12x - 10y = 6
-12x - 9y = -63
Now add vertically
12x - 10y = 6
-12x - 9y = -63
-----------------
-19y = -57
y = -57/(-19)
y = 3
4. Go back to where the terms of the equations are arranged so that
like terms line up vertically like this:
Ax + By = C
Dx + Ey = F
6x - 5y = 3
4x + 3y = 21
2. Get the LCM (least common multiple) of the coefficients of y in
absolute value.
The LCM of 5 and 3 is 15.
3. Multiply each equation by whatever number is necessary to cause
the coefficient of x to be equal to the LCD in absolute value:
The equation 6x - 5y = 3 needs to be multiplied by 3 to make the
coefficient of y have absolute value 15
3[ 6x - 5y = 3]
18x - 15y = 9
The equation 4x + 3y = 21 needs to be multiplied by 5 to make the
coefficient of y to be 15 in absolute value
5[4x + 3y = 21]
20x + 15y = 105
Since these already have their y terms equal in absolute value but
opposite in sign, I don't need to multiply either one through by -1.
So now we have this system:
18x - 15y = 9
20x + 15y = 105
Now add vertically
18x - 15y = 9
20x + 15y = 105
-----------------
38x = 114
y = 114/38
y = 3
------------------
Here are the answers to all your systems:
1. 4x+3y=1, x=1-y ---- answer x=-2, y=3
2. 2x-y=6, -x+y=-1 --- x=5, y=4
3. 6x-y=3, 4x-2y=-2 -- x=1, y=3
4. 2x+3y=7, x=1-4y --- x=5, y=-1
5. 2x+3y=6, x-3y=-15 - x=-3, y=4
6. 7x-5y=4, y=3x-4 --- x= 2, y=2
7. 2y-5x=-1, x=2y+5 -- x=27, y=11
8. 4x+3y=1, 3x+4y=10 - x=-26/7, y = 37/7
9. 6x-5y=3, 4x+3y=21 - x=3, y=3
10. x+y=4, 3x+4y=10 - x=6, y=-2
Edwin
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