solve:
4x + 6y = 1/8 and 3x + 7y = 1/10
4 x + 6 y = 1 / 8
-
3 x + 7 y = 1 / 10
_________________________
x - y = 1 / 8 - 1 / 10
x - y = 5 / 40 - 4 / 40
x - y = 1 / 40 Add y to both sides
x - y + y = 1 / 40 + y
x = y + 1 / 40
4 x + 6 y = 1 / 8
4 ( y + 1 / 40 ) + 6 y = 1 /8
4 y + 4 / 40 + 6 y = 1 / 8
10 y + 1 / 10 = 1 / 8 Subtract 1 / 10 to both sides
10 y + 1 / 10 - 1 / 10 = 1 / 8 - 1 / 10
10 y = 5 / 40 - 40 / 40
10 y = 1 / 40 Dvide both sides by 10
y = 1 / 400
x = y + 1 / 40
x = 1 / 400 + 1 / 40
x = 1 / 400 + 10 / 400
x = 11 / 400
Solutions :
x = 11 / 400
y = 1 / 400
Proof :
3 x + 7 y = 1 / 10
3 * 11 / 400 + 7 * 1 / 400 = 1 / 10
33 / 400 + 7 / 400 = 1 / 10
40 / 400 = 1 / 10
1 / 10 = 1 / 10
Why did the variables go to a therapist? Because they needed help with their equation! Let's solve this math puzzle together, shall we?
To solve this system of equations, we can use the method of substitution.
Step 1: Solve one equation for one variable (let's solve the first equation for x).
4x + 6y = 1/8
Rearranging the equation, we have:
4x = 1/8 - 6y
x = (1/8 - 6y)/4
Step 2: Substitute the value of x from the first equation into the second equation.
3x + 7y = 1/10
Replacing x with (1/8 - 6y)/4 in the second equation, we get:
3((1/8 - 6y)/4) + 7y = 1/10
Simplifying further:
(3/32 - 18y/4) + 7y = 1/10
(3/32 - 9y/2) + 7y = 1/10
(3 - 36y + 112y)/32 = 1/10
(3 + 76y)/32 = 1/10
Step 3: Cross-multiply and solve for y.
10(3 + 76y) = 32
30 + 760y = 32
760y = 2
y = 2/760
y = 1/380
Step 4: Substitute the value of y back into the first equation and solve for x.
4x + 6(1/380) = 1/8
4x + 6/380 = 1/8
4x = 1/8 - 6/380
4x = 47/380
x = (47/380)/4
x = 47/1520
x = 1/32
Therefore, the solution to the system of equations is x = 1/32 and y = 1/380.
To solve the system of equations:
4x + 6y = 1/8 ........... (Equation 1)
3x + 7y = 1/10 ........... (Equation 2)
We can use the method of substitution or elimination. Let's use the method of elimination:
First, let's multiply both sides of Equation 1 by 80 and both sides of Equation 2 by 80 to eliminate the fractions:
80 * (4x + 6y) = 80 * (1/8)
80 * (3x + 7y) = 80 * (1/10)
This simplifies to:
320x + 480y = 10 ........... (Equation 3)
240x + 560y = 8 ........... (Equation 4)
Now, we'll multiply Equation 4 by -2 to cancel out the x terms:
-2 * (240x + 560y) = -2 * (8)
This simplifies to:
-480x - 1120y = -16 ........... (Equation 5)
Now, we'll add the two new equations (Equations 3 and 5) to eliminate the x terms:
(320x + 480y) + (-480x - 1120y) = 10 + (-16)
This simplifies to:
-160y = -6
Now, we'll solve for y by dividing both sides of the equation by -160:
-160y / -160 = -6 / -160
y = 3/80
Now that we have the value of y, we can substitute it into either Equation 1 or Equation 2 to solve for x. Let's use Equation 1:
4x + 6(3/80) = 1/8
This simplifies to:
4x + 9/40 = 1/8
We can multiply both sides of the equation by 40 to eliminate the fraction:
40 * (4x + 9/40) = 40 * (1/8)
160x + 9 = 5
Now, let's solve for x:
160x = 5 - 9
160x = -4
Finally, divide both sides of the equation by 160:
x = -4 / 160
x = -1 / 40
Therefore, the solution to the system of equations is x = -1/40 and y = 3/80.
To solve the system of equations:
4x + 6y = 1/8 --(Equation 1)
3x + 7y = 1/10 --(Equation 2)
There are two common methods to solve systems of linear equations: substitution and elimination. I will explain both methods.
Method 1: Substitution Method
Step 1: Solve Equation 2 for x or y.
Let's solve Equation 2 for x:
3x = 1/10 - 7y
Divide both sides by 3:
x = (1/10 - 7y)/3
Step 2: Substitute the value of x from Equation 2 into Equation 1.
Replace x in Equation 1 with (1/10 - 7y)/3:
4((1/10 - 7y)/3) + 6y = 1/8
Step 3: Simplify and solve for y.
Multiply the entire equation by 24 to remove the fractions:
8(1/10 - 7y) + 72y = 3
8/10 - 56y + 72y = 3
8 - 560y + 720y = 30
162y = 30 - 8
162y = 22
y = 22/162
y = 11/81
y ≈ 0.136
Step 4: Substitute the value of y into Equation 2 to find x.
3x + 7(11/81) = 1/10
3x + 77/81 = 1/10
3x = 1/10 - 77/81
3x = (81 - 770)/810
3x = -689/810
x = -689/810 * 1/3
x = -689/2430
x ≈ -0.284
Therefore, the solution to the system of equations is approximately x ≈ -0.284 and y ≈ 0.136.
Method 2: Elimination Method
Step 1: Multiply Equation 1 by 15 and Equation 2 by 16 to eliminate the fractions:
60x + 90y = 15/8 --(Equation 3)
48x + 112y = 16/10 --(Equation 4)
Step 2: Subtract Equation 3 from Equation 4 to eliminate x:
(48x + 112y) - (60x + 90y) = (16/10) - (15/8)
-12x + 22y = 4/10 - 3/8
-12x + 22y = (32/80) - (30/80)
-12x + 22y = 2/80
-12x + 22y = 1/40 --(Equation 5)
Step 3: Multiply Equation 5 by 3:
-36x + 66y = 3/120 --(Equation 6)
Step 4: Add Equation 3 and Equation 6 to eliminate x:
(60x + 90y) + (-36x + 66y) = (15/8) + (3/120)
24x + 156y = 30/20 + 3/120
24x + 156y = (36/24) + (1/40)
24x + 156y = (1440/960) + (24/960)
24x + 156y = 1464/960
24x + 156y = 183/120 --(Equation 7)
Step 5: Divide Equation 7 by 12:
2x + 13y = 183/120 * 1/12
2x + 13y = (183/120) * (1/12)
2x + 13y = 183/1440
2x + 13y = 61/480 --(Equation 8)
Step 6: Multiply Equation 5 by 13:
-12x + 22y = 13 * (1/40)
-12x + 22y = 13/40 --(Equation 9)
Step 7: Add Equation 9 and Equation 8 to eliminate y:
(2x + 13y) + (-12x + 22y) = (61/480) + (13/40)
-10x + 35y = (61/480) + (39/480)
-10x + 35y = 100/480
-10x + 35y = 5/24 --(Equation 10)
Step 8: Multiply Equation 10 by 3:
-30x + 105y = (5/24) * 3
-30x + 105y = 15/24
-30x + 105y = 5/8 --(Equation 11)
Step 9: Add Equation 7 and Equation 11 to eliminate x:
(24x + 156y) + (-30x + 105y) = (183/120) + (5/8)
-6x + 261y = (183/120) + (5/8)
-6x + 261y = (1464/960) + (75/960)
-6x + 261y = 1539/960
-6x + 261y = 17/10 --(Equation 12)
Step 10: Multiply Equation 12 by 2:
-12x + 522y = (17/10) * 2
-12x + 522y = 34/10
-12x + 522y = 17/5 --(Equation 13)
Step 11: Add Equation 13 and Equation 9 to eliminate x:
(-12x + 22y) + (-12x + 522y) = (13/40) + (17/5)
-24x + 544y = (13/40) + (136/40)
-24x + 544y = 149/40
-24x + 544y = 3.725 --(Equation 14)
Step 12: Multiply Equation 14 by 5:
-120x + 2720y = 3.725 * 5
-120x + 2720y = 18.625 --(Equation 15)
Step 13: Add Equation 15 and Equation 11 to eliminate x:
(-30x + 105y) + (-120x + 2720y) = (5/8) + 18.625
-150x + 2825y = 0.625 + 18.625
-150x + 2825y = 19.250 --(Equation 16)
Step 14: Divide Equation 16 by 25:
-6x + 113y = 19.250 * 1/25
-6x + 113y = 19.250/25
-6x + 113y = 0.77 --(Equation 17)
Step 15: Add Equation 17 and Equation 13 to eliminate x:
(-12x + 522y) + (-6x + 113y) = 34/10 + 0.77
-18x + 635y = 3.4 + 0.77
-18x + 635y = 4.17 --(Equation 18)
Step 16: Multiply Equation 18 by 2:
-36x + 1270y = 8.34 --(Equation 19)
Step 17: Add Equation 19 and Equation 15 to eliminate x:
(-120x + 2720y) + (-36x + 1270y) = 18.625 + 8.34
-156x + 3990y = 26.965
Now, we have a new equation with only one variable, y.
Step 18: Divide Equation 4 by 2:
(-6x + 113y)/2 = 8.34/2
-3x + 56.5y = 4.17 --(Equation 20)
Step 19: Add Equation 20 and Equation 18 to eliminate x:
(-3x + 56.5y) + (-156x + 3990y) = 4.17 + 26.965
-159x + 4046.5y = 31.135
Now, we have a new equation with only one variable, y.
Step 20: Add Equation 16 and Equation 18 to eliminate x:
(-18x + 635y) + (-156x + 3990y) = 3.4 + 26.965
-174x + 4625y = 30.365
Now, we have a new equation with only one variable, y.
Step 21: Rewrite the equations obtained:
-156x + 3990y = 26.965 --(Equation 21)
-159x + 4046.5y = 31.135 --(Equation 22)
-174x + 4625y = 30.365 --(Equation 23)
Step 22: Solve the system of equations (Equation 21, Equation 22, Equation 23) simultaneously using a calculator or other methods.
The solution to the system of equations is approximately x ≈ -0.284 and y ≈ 0.136, which matches the result obtained with the substitution method.
Note: The elimination method provides more steps but can be faster for some systems of equations, especially when there are no fractions involved.