1) -x+10y=60
-4x+16y=32
2) -1/2x - 1/4y=3
1/3x+1/6y=2
#1.
-x + 10y = 60
-4x + 16y = 32
If we need to use the substitution method, we choose an equation and let one of the variable in terms of the other. Then we substitute it to the other equation.
Here, let's choose the first equation.
-x + 10y = 60
Then let's choose the variable x. If we choose x, we must represent it in terms of the other variable (which is y):
-x + 10y = 60
-x = -10y + 60
x = 10y - 60
Finally, substitute it to the second equation:
-4x + 16y = 32
-4(10y - 60) + 16y = 32
Solving for y,
-40y + 240 + 16y = 32
-24y = 32 - 240
-24y = -208
y = 26/3
Substituting to the expression for x:
x = 10(26/3) - 60
x = 260/3 - 180/3
x = 80/3
Now, try doing #2.
Hope this helps :)
Jai - thanks. I tried #1 before you answered and I hot x=32 and y = 6. Wrong??? It worked out in the check
Also, tried #2, getting real lost with the fractions:( Please Help!
It's wrong, I don't know how you checked but x = 32 and y = 6 actually doesn't satisfy either given equation.
Anyway, okay let's do #2.
-1/2x - 1/4y = 3
1/3x + 1/6y = 2
To deal with fractions, it's easier if you first make them whole numbers by multiplying by their LCD (least common denominator).
We multiply the first equation by 4, and we'll get:
-2x - y = 12
We multiply the first equation by 6, and we'll get:
2x + y = 12
Now, if we try adding them,
-2x - y = 12
2x + y = 12
----------------------
0 + 0 = 24
0 = 24
which is absolutely impossible. Therefore, no solutions exist. (If you plot them, they are actually parallel to each other, and indeed, no solutions exist as they will never intersect)
Hope this helps :3
Jai - it was -5x+10y=60. Typo - wow so sorry - but now was my answer right?
To solve these systems of equations, you can use either the substitution method or the elimination method. I will explain how to solve each system of equations using both methods.
1)
Substitution Method:
Step 1: Solve one equation for one variable in terms of the other variable. Let's solve the first equation for x in terms of y:
-x + 10y = 60
-x = 60 - 10y
x = -60 + 10y
Step 2: Substitute the expression for x into the other equation:
-4(-60 + 10y) + 16y = 32
240 - 40y + 16y = 32
-24y = -208
y = -208 / -24
y = 8.67 (rounded to two decimal places)
Step 3: Substitute the value of y back into one of the original equations to solve for x:
-x + 10(8.67) = 60
-x + 86.7 = 60
-x = -26.7
x = 26.7
So, the solution to the system of equations is x = 26.7 and y = 8.67.
Elimination Method:
Step 1: Multiply one or both equations by constants to make the coefficients of one of the variables the same or opposite.
Let's multiply the first equation by 4 to make the coefficients of x the same:
-4x + 40y = 240
Step 2: Add or subtract the equations to eliminate one of the variables. In this case, we'll subtract the second equation from the first:
(-4x + 40y) - (-4x + 16y) = 240 - 32
-4x + 40y + 4x - 16y = 208
24y = 208
y = 8.67 (rounded to two decimal places)
Step 3: Substitute the value of y back into one of the original equations to solve for x:
-x + 10(8.67) = 60
-x + 86.7 = 60
-x = -26.7
x = 26.7
The solution is x = 26.7 and y = 8.67.
2)
Substitution Method:
Step 1: Solve one equation for one variable in terms of the other variable. Let's solve the first equation for x in terms of y:
-1/2x - 1/4y = 3
-1/2x = 3 + 1/4y
x = -6 - 1/2y
Step 2: Substitute the expression for x into the other equation:
1/3(-6 - 1/2y) + 1/6y = 2
-2 -1/6y + 1/6y = 2
-2 = 2
Since the equation is inconsistent (the left side does not equal the right side), this system of equations has no solution.
Elimination Method:
Step 1: Multiply one or both equations by constants to make the coefficients of one of the variables the same or opposite.
In this case, let's multiply the second equation by 2 to make the coefficients of y the same:
2(1/3x + 1/6y) = 2(2)
2/3x + 1/3y = 4
Step 2: Add or subtract the equations to eliminate one of the variables. In this case, we'll subtract the first equation from the second:
(2/3x + 1/3y) - (-1/2x - 1/4y) = 4 - 3
2/3x + 1/3y + 1/2x + 1/4y = 1
4/6x + 2/6x + 2/6y + 1/4y = 1
10/6x + 10/6y = 1
5/3x + 5/3y = 1
The resulting equation still contains x and y, and it is not possible to isolate either variable. Therefore, this system of equations has no solution.
In summary, the first system of equations has a solution of x = 26.7 and y = 8.67, while the second system of equations has no solution.