Solve for x and y:
2x + y = 1
x - y = 8
PLEASE HELP. and show work.(:
In a set of simultaneous equations, just add these two equations so as to cancel one of the terms. In this case that is easy to do; if they don't automatically add to cancel one of the variables, then multiply one or both of the equations to make them cancel.
2x+y=1
x-y=8
=============
adding we obtain
3x=9
x = 3
Now substitute the 3 for x into either of the equations; e.g.
x-y=8
3-y=8
-y=8-3=5
y=-5
Now check your solution to see that x = 3 and y=-5 will satisfy BOTH equations.
x-y=8
3-(-5)=8
3+5=8
8=8
======
Now the other equation.
2x+y=1
2(3)+(-5)=1
6-5=1
1=1
both check
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But what if one of the variables won't cancel. Here is what you can do.
2x+y=1
x-y=8
=========
Now multiply equation 1 by -1; follow that by multiplying equation 2 by 2.
-x+y=-8 and
-2x+2y=-16
Now add in equation 1 like so
-2x+2y=-16
2x+y=1
=============
3y=-15
y=-5 which is the answer we obtained above. Now substitute -5 into either equation and solve for x.
I hope this helps.
To solve the system of equations:
Step 1: Begin by isolating one variable in one of the equations. Let's isolate x in the second equation:
x - y = 8
Add y to both sides:
x = 8 + y
Step 2: Substitute the value of x from the second equation into the first equation:
2x + y = 1
Replace x with 8 + y:
2(8 + y) + y = 1
Distribute 2 to both terms inside the parentheses:
16 + 2y + y = 1
Combine like terms:
3y + 16 = 1
Step 3: Solve for y. First, subtract 16 from both sides:
3y = 1 - 16
3y = -15
Divide both sides by 3:
y = -15/3
y = -5
Step 4: Substitute the value of y back into the second equation to find x:
x - (-5) = 8
Simplify:
x + 5 = 8
Subtract 5 from both sides:
x = 8 - 5
x = 3
Step 5: Check the solution by substituting the values of x and y into the first equation:
2x + y = 1
Replace x with 3 and y with -5:
2(3) + (-5) = 1
6 - 5 = 1
1 = 1
Since both equations are satisfied by the values x = 3 and y = -5, the solution is x = 3 and y = -5.
To solve for x and y, we can use the method of substitution or elimination. Let's use the method of elimination in this case:
Given equations:
1) 2x + y = 1
2) x - y = 8
Step 1: Multiply equation 2 by 2 to eliminate the variable x:
2 * (x - y) = 2 * 8
2x - 2y = 16
Now we have the following equations:
1) 2x + y = 1
2) 2x - 2y = 16
Step 2: Add equations 1 and 2 to eliminate the variable x:
(2x + y) + (2x - 2y) = 1 + 16
4x - y = 17
Now we have the following equation:
3) 4x - y = 17
Step 3: Now we have a system of two equations with two variables:
2x + y = 1
4x - y = 17
To eliminate the variable y, add equations 1 and 3:
(2x + y) + (4x - y) = 1 + 17
6x = 18
Divide both sides of the equation by 6:
x = 18 / 6
x = 3
Step 4: Substitute the value of x back into equation 1 to solve for y:
2x + y = 1
2(3) + y = 1
6 + y = 1
Subtract 6 from both sides of the equation:
y = 1 - 6
y = -5
Therefore, the solution to the system of equations is x = 3 and y = -5.