I have to solve this equation using substitution.
3x+5y=10
2x+ 1/2y=24
This is all one problem
2 x + ( 1 / 2 ) y = 24 Multiply both sides by 10
20 x + ( 1 / 2 ) y * 10 = 24 * 10
20 x + 5 y = 240
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3 x + 5 y = 10
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20 x + 5 y = 240
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3 x - 20 x + 5 y - 5 y = 10 - 240
- 17 x = - 230 Divide both sides by - 17
- 17 x / - 17 = - 230 / - 17
x = 230 / 17
2 x + 1 / 2 y = 24
2 * 230 / 17 + ( 1 / 2 ) y = 24
460 / 17 + ( 1 / 2 ) y = 24 Subtract 460 / 17 to both sides
460 / 17 + ( 1 / 2 ) y - 460 / 17 = 24 - 460 / 17
( 1 / 2 ) y = 24 - 460 / 17
( 1 / 2 ) y = 24 * 17 / 17 - 460 / 17
( 1 / 2 ) y = 408 / 17 - 460 / 17
( 1 / 2 ) y = - 52 / 17 Multiply both sides by 2
y = - 104 / 17
Solutions :
x = 230 / 17 , y = - 104 / 17
using substitution:
multiply the 2nd by 2
4x + y = 48
y = 48-4x
sub into the 1st
3x + 5(48-4x) = 10
3x + 240 - 20x = 10
-17x = -230
x = 230/17
then y = 48-4(230/7) = -104/17
To solve the given system of equations using substitution, we will first solve one equation for one variable and substitute that expression into the other equation.
Let's start with the first equation and solve it for x:
3x + 5y = 10
Rearrange the equation to isolate x:
3x = 10 - 5y
Divide both sides of the equation by 3:
x = (10 - 5y) / 3
Now, substitute this expression for x into the second equation:
2x + (1/2)y = 24
Replace x with (10 - 5y) / 3:
2((10 - 5y) / 3) + (1/2)y = 24
To simplify the equation, distribute 2 to (10 - 5y):
(20 - 10y) / 3 + (1/2)y = 24
Next, we need to clear the fractions by multiplying every term by 6 to eliminate the denominators (3 and 2):
6((20 - 10y) / 3) + 6((1/2)y) = 6(24)
This simplifies to:
2(20 - 10y) + 3y = 144
Distribute 2 to (20 - 10y):
40 - 20y + 3y = 144
Combine like terms:
40 - 17y = 144
Now, isolate y by subtracting 40 from both sides:
-17y = 144 - 40
Simplify:
-17y = 104
Divide both sides by -17 to solve for y:
y = 104 / -17
y ≈ -6.12 (rounded to two decimal places)
Now that we have the value of y, substitute it back into the expression for x:
x = (10 - 5y) / 3
x = (10 - 5(-6.12)) / 3
Calculate:
x ≈ 11.04
So the solution to the system of equations is approximately x ≈ 11.04 and y ≈ -6.12.