How would you solve these systems by elimination? I'm in desperate need of help!
1. -35x-15y=0
-56x+24y=0
2. -56x+72y=-16
-49x+63y=-14
we must find a common multiple of something here
15 = 3*5
24 = 3*8
so I guess we could use 3*5*8 = 120
15 * 8 = 120 so multiply first eqn by 8
24 * 5 = 120 so multiply second by 5
- 280 x - 120 y = 0
- 200 x + 120 y = 0
-------------------- add
-480 x = 0
so x = 0
and y = 0
which we could have said back at the beginning if we had sketched graphs
in the second one both 56 and 49 have common factor 7 which we will take advantage of
56 = 7 * 8
49 = 7*7 so use 7*7*8 = 392
56 *7 = 392 so multiply first by 7
49 *8 = 392 so multiply second times 8
I think you can take it from there.
On the first one how did you get -280 and -200?
-280 = -35 * 8
-280 =-56 * 5
sorry, 280 not 200, does not matter, solution is origin (0,0) anyway
both of those lines are through the origin having no b in y = m x + b
To solve the systems by elimination, follow these steps:
Step 1: Choose an equation and multiply it by a constant to make the coefficients of one of the variables in the two equations equal or additive inverses of each other.
Step 2: Add or subtract the equations to eliminate one variable.
Step 3: Solve the resulting equation for the remaining variable.
Step 4: Substitute the value of the solved variable back into one of the original equations and solve for the other variable.
Step 5: Write the solution as an ordered pair (x, y).
Let's apply these steps to the systems you provided:
1. -35x - 15y = 0
-56x + 24y = 0
Step 1: Multiply the first equation by -8 and the second equation by 5 to make the coefficients of x equal additive inverses.
-35x - 15y = 0 (multiply by -8)
280x + 120y = 0
-56x + 24y = 0 (multiply by 5)
-280x + 120y = 0
The equations become:
280x + 120y = 0
-280x + 120y = 0
Step 2: Add the equations to eliminate x.
(280x + 120y) + (-280x + 120y) = 0 + 0
240y = 0
So we are left with 240y = 0.
Step 3: Solve for y.
240y = 0
y = 0/240
y = 0
Step 4: Substitute the value of y back into one of the original equations.
-35x - 15y = 0
-35x - 15(0) = 0
-35x = 0
x = 0/-35
x = 0
Step 5: Write the solution as an ordered pair (x, y).
The solution to the system is (0, 0).
Now let's solve the second system:
2. -56x + 72y = -16
-49x + 63y = -14
Step 1: Multiply the first equation by -7 and the second equation by 8 to make the coefficients of x equal additive inverses.
-56x + 72y = -16 (multiply by -7)
392x - 504y = 112
-49x + 63y = -14 (multiply by 8)
-392x + 504y = -112
The equations become:
392x - 504y = 112
-392x + 504y = -112
Step 2: Add the equations to eliminate x.
(392x - 504y) + (-392x + 504y) = 112 + (-112)
0 = 0
The resulting equation is 0 = 0, which means the variables have been eliminated, and the system is called dependent or consistent.
This means that the system has infinitely many solutions. Any values of x and y that satisfy one equation will satisfy the other equation.
There is no single unique solution for this system. Instead, you can express the solution as a set of ordered pairs or as a formula in terms of y.
I hope this explanation helps you understand how to solve systems by elimination!