Hi, I need help on this problem:Solve the system of equation by giving an ordered pair for 14x +3y = 115
7x +3y = 80
Use the 3 Methods in your work.
Thank You, I am not sure how to use graphing, substitution, and elimination, please show these methods.
Sure! I'll explain each method and then show you how to use them to solve the system of equations.
Method 1: Graphing
To solve a system of equations using graphing, you need to graph both equations on the same coordinate plane and find the point where the two lines intersect. The coordinates of this point will be the solution to the system.
Method 2: Substitution
In substitution, you solve one equation for one variable and then substitute that expression into the other equation. By doing this, you can solve for the remaining variable. Once you have the value of one variable, you can substitute it into either of the original equations to find the value of the other variable.
Method 3: Elimination
In elimination, you manipulate the equations in a way that allows you to add or subtract them to eliminate one variable. This creates a new equation with only one variable, which can then be solved. Once you obtain the value of one variable, you can substitute it back into one of the original equations to find the value of the second variable.
Let's solve the system of equations using these three methods:
Method 1: Graphing
To start, we can rearrange the equations to solve for y:
Equation 1: 14x + 3y = 115
Equation 2: 7x + 3y = 80
Now, graph both of these equations on the coordinate plane. The point where the two lines intersect will give us the solution. If the lines are parallel, it means there is no solution, and if the lines are coincident, it means there are infinitely many solutions.
Method 2: Substitution
Let's solve for y in Equation 1:
14x + 3y = 115
3y = 115 - 14x
y = (115 -14x)/3
Now we substitute this expression for y into Equation 2:
7x + 3(115 - 14x)/3 = 80
Simplify the equation and solve for x:
7x + 115 - 14x = 80
-7x = 80 - 115
-7x = -35
x = -35 / -7
x = 5
Substitute the value of x back into Equation 1:
14(5) + 3y = 115
70 + 3y = 115
3y = 115 - 70
3y = 45
y = 45 / 3
y = 15
So, the ordered pair (x, y) is (5, 15) which is the solution to the system of equations.
Method 3: Elimination
First, let's multiply Equation 2 by 2 to get rid of the y variable.
Equation 1: 14x + 3y = 115
Equation 2: 14x + 6y = 160
Now we subtract Equation 1 from Equation 2 to eliminate the x variable:
(14x + 6y) - (14x + 3y) = 160 - 115
3y = 45
y = 45 / 3
y = 15
Substitute the value of y back into Equation 1:
14x + 3(15) = 115
14x + 45 = 115
14x = 115 - 45
14x = 70
x = 70 / 14
x = 5
Again, we get the ordered pair (x, y) as (5, 15) which is the solution to the system of equations using the elimination method.
I hope this helps! Let me know if you have any further questions.
elimination:
subtract the 2nd from the first (you have 3y in both)
7x = 35
x = 5
sub into the 2nd:
7(5) + 3y = 80
3y = 45
y = 15
They intersect at (5,15)
substitution: from the 2nd :
3y = 80-7x
plug into the first,
14x + 80-7x = 115
7x = 35
x = 5
then 3y = 80-7x = 80 - 35
3y = 45
y = 15 , same as above
graphing
http://www.wolframalpha.com/input/?i=plot+14x+%2B3y+%3D+115+%2C+7x+%2B3y+%3D+80
I assume you know how to graph a straight line