Solve the system of two liner equations -3x-4y=-23 and 3x-5y=5
To solve the system of linear equations, we can use either the substitution method or the elimination method. I will use the elimination method.
First, let's multiply the second equation by 2 to eliminate x:
2*(3x - 5y) = 2*5
6x - 10y = 10
Now we have the system of equations:
-3x - 4y = -23 (Equation 1)
6x - 10y = 10 (Equation 2)
Next, add Equation 1 and Equation 2:
(-3x - 4y) + (6x - 10y) = (-23) + (10)
3x - 14y = -13 (Equation 3)
Now we have the system of equations:
-3x - 4y = -23 (Equation 1)
3x - 14y = -13 (Equation 3)
To eliminate x, multiply Equation 1 by 3 and Equation 3 by -3:
3*(-3x - 4y) = 3*(-23)
-9x - 12y = -69 (Equation 4)
-3*(3x - 14y) = -3*(-13)
-9x + 42y = 39 (Equation 5)
Now we have the system of equations:
-9x - 12y = -69 (Equation 4)
-9x + 42y = 39 (Equation 5)
Next, add Equation 4 and Equation 5:
(-9x - 12y) + (-9x + 42y) = (-69) + (39)
-18x + 30y = -30
Simplify the equation:
-6x + 10y = -10 (Equation 6)
So, now we have the system of equations:
-6x + 10y = -10 (Equation 6)
3x - 14y = -13 (Equation 3)
Now, let's multiply Equation 6 by -5:
-5*(-6x + 10y) = -5*(-10)
30x - 50y = 50 (Equation 7)
Now we have the system of equations:
30x - 50y = 50 (Equation 7)
3x - 14y = -13 (Equation 3)
Next, add Equation 7 and Equation 3:
(30x - 50y) + (3x - 14y) = 50 + (-13)
33x - 64y = 37
So now we have the system of equations:
33x - 64y = 37 (Equation 8)
3x - 14y = -13 (Equation 3)
Now, we can solve this system of equations. Subtract Equation 3 multiplied by 11 from Equation 8:
33x - 64y - (3x - 14y) * 11 = 37 - (-13) * 11
33x - 64y - 33x + 154y = 37 + 143
-64y + 154y = 37 + 143
90y = 180
y = 180/90
y = 2
Now substitute y = 2 into Equation 3 and solve for x:
3x - 14*2 = -13
3x - 28 = -13
3x = -13 + 28
3x = 15
x = 15/3
x = 5
Therefore, the solution to the system of equations -3x - 4y = -23 and 3x - 5y = 5 is x = 5 and y = 2.
To solve the system of linear equations:
Equation 1: -3x - 4y = -23
Equation 2: 3x - 5y = 5
We can solve this system by using the method of elimination.
Step 1: Multiply Equation 1 by 3 and Equation 2 by -3 to eliminate the variable x.
-3x - 4y = -23 (Equation 1)
3x - 5y = 5 (Equation 2)
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-9x - 12y = -69 (Multiply Equation 1 by 3)
-9x + 15y = -15 (Multiply Equation 2 by -3)
Step 2: Add the two new equations to eliminate x.
(-9x - 12y) + (-9x + 15y) = (-69) + (-15)
-18x + 3y = -84
Step 3: Simplify the equation.
-18x + 3y = -84
Step 4: Divide all terms by -3 to solve for y.
(-18x + 3y)/-3 = -84/-3
6x - y = 28
Step 5: Rearrange the equation.
-y = -6x + 28
Step 6: Multiply all terms by -1 to solve for y.
(-y)(-1) = (-6x + 28)(-1)
y = 6x - 28
Step 7: Substitute the value of y in terms of x into either Equation 1 or Equation 2 to solve for x. We'll use Equation 1.
-3x - 4(6x - 28) = -23
-3x - 24x + 112 = -23
-27x + 112 = -23
Step 8: Subtract 112 from both sides of the equation.
-27x + 112 - 112 = -23 - 112
-27x = -135
Step 9: Divide all terms by -27 to solve for x.
(-27x)/-27 = (-135)/-27
x = 5
Step 10: Substitute the value of x back into Equation 1 to solve for y.
-3(5) - 4y = -23
-15 - 4y = -23
Step 11: Add 15 to both sides of the equation.
-15 - 4y + 15 = -23 + 15
-4y = -8
Step 12: Divide all terms by -4 to solve for y.
(-4y)/-4 = (-8)/-4
y = 2
Therefore, the solution to the system of linear equations is x = 5 and y = 2.
To solve the system of linear equations:
-3x - 4y = -23
3x - 5y = 5
We can use the method of substitution or elimination. Let's use the method of elimination.
Step 1: Multiply the second equation by 4 and the first equation by 5 to eliminate the x term:
4(3x - 5y) = 4(5)
5(-3x - 4y) = 5(-23)
12x - 20y = 20
-15x - 20y = -115
Step 2: Subtract the second equation from the first equation to eliminate the y term:
(12x - 20y) - (-15x - 20y) = 20 - (-115)
12x - 20y + 15x + 20y = 20 + 115
27x = 135
x = 135 / 27
x = 5
Step 3: Substitute the value of x back into one of the original equations to solve for y. Let's use the first equation:
-3x - 4y = -23
-3(5) - 4y = -23
-15 - 4y = -23
-4y = -23 + 15
-4y = -8
y = -8 / -4
y = 2
Therefore, the solution to the system of linear equations is x = 5 and y = 2.
To verify the solution, substitute these values back into both equations and check that they hold true.