Solve the system of equations.
3x+6y=21
–5x+2y=13
To solve the system of equations, we can use the method of substitution or elimination.
Using the method of substitution:
From equation 1, we can isolate x:
3x + 6y = 21
3x = 21 - 6y
x = (21 - 6y)/3
x = 7 - 2y
Substituting this expression for x into equation 2:
-5(7 - 2y) + 2y = 13
-35 + 10y + 2y = 13
12y = 48
y = 48/12
y = 4
Substituting the value of y back into the expression for x:
x = 7 - 2(4)
x = 7 - 8
x = -1
Therefore, the solution to the system of equations is x = -1 and y = 4.
To solve the system of equations:
Step 1: Multiply the first equation by 5 and the second equation by 3 to make the coefficients of the x terms the same.
5(3x + 6y) = 5(21)
3(-5x + 2y) = 3(13)
Simplifying these equations, we get:
15x + 30y = 105
-15x + 6y = 39
Step 2: Add both equations together to eliminate the x term.
(15x + 30y) + (-15x + 6y) = 105 + 39
Simplifying, we have:
36y = 144
Step 3: Divide both sides of the equation by 36 to solve for y.
36y/36 = 144/36
y = 4
Step 4: Substitute the value of y back into one of the original equations to solve for x. Let's use the first equation.
3x + 6(4) = 21
3x + 24 = 21
3x = 21 - 24
3x = -3
x = -1
So the solution to the system of equations is x = -1 and y = 4.
To solve the system of equations, we can use either the substitution method or the elimination method. I will explain how to solve it using the elimination method.
Step 1: Multiply both sides of the second equation by 3 to make the coefficient of x in both equations equal.
Original equations:
3x + 6y = 21
-5x + 2y = 13
Multiply the second equation by 3:
-15x + 6y = 39
Step 2: Add the two equations together to eliminate x.
(3x + 6y) + (-15x + 6y) = 21 + 39
Simplifying the equation:
-12x + 12y = 60
Step 3: Divide both sides of the equation by 12 to solve for y.
(1/12)(-12x + 12y) = (1/12)(60)
Simplifying:
-x + y = 5
Step 4: Now we have a new equation. We can substitute this equation into one of the original equations to solve for x or y. Let's use the first equation.
3x + 6y = 21
Substitute for y:
3x + 6(5 - x) = 21
Simplifying:
3x + 30 - 6x = 21
-3x + 30 = 21
-3x = 21 - 30
-3x = -9
Step 5: Divide both sides of the equation by -3 to solve for x.
(-1/3)(-3x) = (-1/3)(-9)
Simplifying:
x = 3
Step 6: Substitute the value of x into one of the original equations to solve for y. Let's use the first equation.
3x + 6y = 21
Substitute for x:
3(3) + 6y = 21
Simplifying:
9 + 6y = 21
6y = 21 - 9
6y = 12
Step 7: Divide both sides of the equation by 6 to solve for y.
(1/6)(6y) = (1/6)(12)
Simplifying:
y = 2
Therefore, the solution to the system of equations is x = 3 and y = 2.