Solve: 3x-2y=10 2x+3y=-2
Graph:
Substitution:
Elimination:
Show Your work!
(I can'tfigure out how to graph it and when i used substitution and elimination it came out with two different anwsers. I need help! and i need to be able to show my work!)
Eq1: 3X - 2Y = 10.
Eq2: 2X + 3Y = -2.
Substitution Method:
In Eq2, solve for X:
Eq2: 2X + 3Y = -2,
2X = -3Y - 2,
X = -3Y/2 - 1,
In Eq1, substitute -3Y/2 - 1 for X:
3(-3X/2 - 1) - 2Y = 10,
-9X/2 - 3 - 2Y = 10,
Multiply both sides by 2 and get:
-9X - 6 -4Y = 20,
-13Y = 20 + 6,
-13Y = 26,
Y = 26 / -13 = - 2.
In Eq2, substitute -2 for Y:
2X + 3*-2 = -2,
2X - 6 = - 2,
2X = -2 + 6,
2X = 4,
X = 4 / 2 = 2.
Solution Set = (2, -2).
Elimination Method:
Multiply Eq1 by -2 and Eq2 by 3 and get:
-6X + 4Y = -20,
+6X + 9Y = -6
Add the 2 Eqs and get:
13Y = -26,
Y = -26 / 13 = -2.
Substitute -2 for Y in Eq1:
3X - 2*-2 = 10,
3X + 4 = 10,
3X = 10 - 4,
3X = 6,
X = 6 / 3 = 2.
Solution Set = (2, -2).
GRAPH: Use the following points.
Eq1:(-2, -8) (0, -5), (2, -2).
Eq2: (-2.5, 1),(-4, 2) (2, -2).
The point where the lines intersect is the solution.
To solve the system of equations 3x - 2y = 10 and 2x + 3y = -2, we can use graphing, substitution, or elimination methods. Let's go through each method step by step and show the work.
1. Graphing Method:
To graph the system of equations, we can plot the points where the two lines intersect.
First, rearrange both equations in slope-intercept form (y = mx + b):
Equation 1: 3x - 2y = 10
-2y = -3x + 10
y = (3/2)x - 5
Equation 2: 2x + 3y = -2
3y = -2x - 2
y = (-2/3)x - (2/3)
Now, we can graph these two lines on a coordinate plane. Since the graphs intersect at a single point, that point represents the solution to the system.
2. Substitution Method:
To use substitution, solve one equation for one variable and substitute that expression into the other equation.
Let's solve the first equation, 3x - 2y = 10, for x:
3x = 2y + 10
x = (2y + 10)/3
Now, substitute this expression for x in the second equation, 2x + 3y = -2:
2((2y + 10)/3) + 3y = -2
(4y + 20)/3 + 3y = -2
Multiply through by 3 to eliminate the denominator:
4y + 20 + 9y = -6
13y = -26
y = -2
Substitute this value of y back into the first equation to solve for x:
3x - 2(-2) = 10
3x + 4 = 10
3x = 6
x = 2
So, the solution is x = 2 and y = -2.
3. Elimination Method:
To use elimination, multiply both equations to make the coefficients of one variable in both equations equal but opposite.
Multiply the first equation, 3x - 2y = 10, by 3 and the second equation, 2x + 3y = -2, by 2:
3(3x - 2y) = 3(10) --> 9x - 6y = 30
2(2x + 3y) = 2(-2) --> 4x + 6y = -4
Now, add the two equations together to eliminate the y variable:
9x - 6y + 4x + 6y = 30 + (-4)
13x = 26
x = 2
Substitute this value of x back into one of the original equations to solve for y. Let's use the first equation:
3(2) - 2y = 10
6 - 2y = 10
-2y = 4
y = -2
Again, the solution is x = 2 and y = -2.
All three methods yield the same result, x = 2 and y = -2, as the solution to the given system of equations.