S = {-6,4}, x = -6 y = 4
Step-by-step explanation:
the substitution method is when you solve one of the equations for a variable, then plug in that result into the other equation and solve
the addition/elimination method is when you get both equations to have the same value for a variable and 'eliminate' it and solve for the other variable
i personally prefer the substitution method as it allows us to graph easier, but i will demonstrate both methods
SUBSTITUTION METHOD:
to start off, we will solve for one variable in one of the equation. i have chosen to solve for x in 3x + 6y = 6
3x + 6y = 6 < subtract 6y from both sides
3x = -6y + 6 < divide both sides by 3 to isolate x
3x/3 = x
-6y/3 = -2y
6/3 = 2
x = -2y + 2 or x = 2 - 2y
now we will plug in the x value we have found into 5x + 4y = -14
solve: 5x + 4y = -14 when x = 2 - 2y
5(2 - 2y) + 4y = -14 < distribute 5 into 2 - 2y
10 - 10y + 4y = -14 < combine like terms
-10y + 4y = -6y
10 - 6y = -14 < now we solve for y: subtract 10 from both sides
-6y = -24 < divide both sides by -6 to isolate y
-6y/-6 = y
-24/-6 = 4
y = 4
now that we have found y, we can plug this y value in to find x in the system. you can choose any equation you want, i will plug the y value into 3x + 6y = 6
solve: 3x + 6y = 6 when y = 4
3x + 6(4) = 6
3x + 24 = 6 < subtract 24 from both sides
3x = -18 < divide both sides by 3
3x/3 = x
-18/3 = -6
x = -6
our answer to this system of equation is {-6,4}
i will now solve this same system with the addition/elimination method:
ADDITION/ELIMINATION METHOD:
to solve, we need to 'eliminate' one of the variable terms in both equations
5x + 4y = -14
3x + 6y = 6
i will eliminate the x value in this instance, but you can eliminate either variable. to eliminate, we need one negative number and one positive so they cancel out. both 5 & 3 have the number 15 as a common factor, we so we can multiply each number to get 15 in one equation and -15 in another.
-3(5x + 4y = -14)
5(3x + 6y = 6)
we absolutely must multiply the entire equation by the number to get the correct answer. we distribute -3 into 5x + 4y = -14 and 5 into 3x + 6y = 6
-15x - 12y = 42
15x + 30y = 30
add both equations above and -15x & 15x will cancel, thus eliminating the x term
-15x + 15x = 0
-12y + 30y = 18y
42 + 30 = 72
our new equation is:
18y = 72 < divide both sides by 18 to isolate y
18y/18 = y
72/18 = 4
y = 4
now we plug in the y value we have found into one of the equations above. i will plug in y = 4 into 3x + 6y = 6
solve: 3x + 6y = 6 when y = 4
3x + 6(4) = 6
3x + 24 = 6 < subtract 24 from both sides
3x = -18 < divide both sides by 3
3x/3 = x
-18/3 = -6
x = -6
out solution to this system of inequalities is {-6,4}
the solution is the same with both the elimination & substitution method, so you can use any method you want to solve.
please let me know if i need to be more clear!
