13×+22y = 1610
32/× +12y = 22/10
To solve this system of equations, we can use the method of substitution.
From the second equation, we can rewrite it as:
32/x + 12y = 2.2
Next, we can multiply both sides of this equation by x to get rid of the fraction:
32 + 12xy = 2.2x
Now, we can rearrange this equation to solve for x:
2.2x - 12xy = 32
Factor out x:
x(2.2 - 12y) = 32
Divide both sides by (2.2 - 12y) to solve for x:
x = 32 / (2.2 - 12y)
Now, we can substitute this expression for x in the first equation:
13(32 / (2.2 - 12y)) + 22y = 1610
Multiply both sides of this equation by (2.2 - 12y) to eliminate the denominator:
13(32) + 22y(2.2 - 12y) = 1610(2.2 - 12y)
Simplify:
416 + 22y(2.2 - 12y) = 3542 - 19320y
Expand:
416 + 48.4y - 264y^2 = 3542 - 19320y
Rearrange this equation to form a quadratic equation:
264y^2 - 48.4y - 19320y + 3542 - 416 = 0
Combine like terms:
264y^2 - 19632y + 3126 = 0
Divide through the equation by 6 to simplify:
44y^2 - 3272y + 521 = 0
This is a quadratic equation that we can solve for y using factoring, completing the square, or the quadratic formula.
To solve the system of equations:
Equation 1: 13x + 22y = 1610
Equation 2: (32/x) + 12y = 22/10
Step 1: Simplify Equation 2 by multiplying both sides by 10x:
10x * ((32/x) + 12y) = 10x * (22/10)
320 + 120xy = 22x
Step 2: Rearrange Equation 1 by isolating the x term:
13x = 1610 - 22y
13x = 1610 - 22y
Step 3: Substitute the value of 13x from Equation 1 into Equation 2:
320 + 120xy = 22(1610 - 22y)
Step 4: Simplify the equation:
320 + 120xy = 35420 - 484y
Step 5: Rearrange the equation to isolate the y term:
484y - 120xy = 35420 - 320
Step 6: Combine like terms:
484y - 120xy = 35100
Step 7: Factor out y:
y(484 - 120x) = 35100
Step 8: Solve for y by dividing both sides by (484 - 120x):
y = 35100 / (484 - 120x)
Step 9: Substitute the obtained value of y into Equation 1 to solve for x:
13x + 22(35100 / (484 - 120x)) = 1610
Step 10: Simplify the equation and solve for x.
To solve the system of equations:
13x + 22y = 1610 --- Equation (1)
32/x + 12y = 22/10 --- Equation (2)
We can start by isolating a variable in one of the equations, and then substitute it into the other equation. Let's isolate "x" in Equation (2):
32/x + 12y = 22/10
To get rid of the fraction, we can multiply both sides of the equation by "x":
x * (32/x) + 12yx = (22/10) * x
32 + 12xy = 22x/10
Next, let's simplify the fraction on the right side:
32 + 12xy = (11/5) * x
Now, let's rearrange the equation:
12xy - (11/5)x = -32
Factor out "x":
x(12y - 11/5) = -32
Now we have an expression for "x" in terms of "y". To solve for "y", we can substitute this expression into Equation (1):
13x + 22y = 1610
Replace "x" with the expression we obtained:
13(12y - 11/5) + 22y = 1610
Distribute:
156y - 143/5 + 22y = 1610
Combine like terms:
178y - 143/5 = 1610
Add 143/5 to both sides:
178y = 1610 + 143/5
Convert 1610 to a fraction with a common denominator of 5:
178y = 8050/5 + 143/5
Combine the fractions:
178y = 8193/5
Divide by 178:
y = (8193/5) / 178
Simplify the fraction:
y = 8193/890
Now that we have the value of "y", we can substitute it back into Equation (1) to solve for "x":
13x + 22(8193/890) = 1610
Multiply 22 with (8193/890):
13x + (180426/890) = 1610
Subtract (180426/890) from both sides:
13x = 1610 - (180426/890)
Combine the fractions on the right side:
13x = (1610 * 890 - 180426) / 890
Calculate the numerator:
13x = 1429900 - 180426
13x = 1249474
Divide by 13:
x = 1249474 / 13
Simplify the fraction:
x = 96190
Therefore, the solution to the system of equations is x = 96190 and y = 8193/890.