1. Verify that (-4, 12) is the solution to the system.
Show work to justify your answer
2x+y=5
-5x-2y=-6
2. Solve the system by graphing. State the solution.
x + y =2
2y – x = 10
3. Solve the system by substitution. State whether the
system has one solution, infinite solutions, or no
solution.
2x + y = 10
x – 3y = -2
Can someone please explain to me how to do these?
1.
substitute -4 for x and substitute 12 for y
2(-4) + 12 = 5 ??? NO WAY !!!!!
2. sketch graph
line one goes down from 2 on x axis at 45 degrees to right
line two goes from x = -10 when y = 0 up at 30 degrees to right
they hit at x = -2, y = 4
Thank you guys SO MUCH
from first
y = (10-2x)
use in second
x - 3(10-2x) = -2
x - 30 + 6 x = -2
7 x = 28
x = 28/4 = 7
now go back and get y
1. plug -4 and 12 into both equations to confirm the solution
2. graph the two equations...the solution is the intersection of the lines
... there are several graphing programs online ... or you could use graph paper, ruler, and pencil
3. solve the 2nd equation for x, in terms of y
... substitute the value for x into the 1st equation
... solve for y, then solve for x
1. Well, if we want to verify if (-4, 12) is the solution to the system, we can substitute these values into the equations and see if they hold true.
For the first equation, 2x + y = 5, when we substitute x = -4 and y = 12, we get:
2(-4) + 12 = 5
-8 + 12 = 5
4 = 5
Oops! It seems that when we substituted these values, the equation is not true. So (-4, 12) is NOT the solution to the system.
For the second equation, -5x - 2y = -6, we can substitute the same values and check if it's true:
-5(-4) - 2(12) = -6
20 - 24 = -6
-4 = -6
Again, the equation is not true. So (-4, 12) is NOT the solution to the system.
2. To solve the system by graphing, plot the lines represented by each equation on the same graph. The point where the lines intersect will be the solution to the system.
For the first equation, x + y = 2, we can rearrange it to y = -x + 2.
For the second equation, 2y – x = 10, we can rearrange it to y = (1/2)x + 5.
Now, let's plot these lines:
|
5 |
|
4 |
| .
3 |
______|_____|_____
2 | .
|
1 |
|
0 |
-1 |
|
-2 |
|
-3 |
|
-4 X|
_______________
The lines intersect at the point (3, -1), so the solution to the system is x = 3 and y = -1.
3. To solve the system by substitution, we need to solve one equation for one variable and substitute it into the other equation.
Let's start by solving the first equation, 2x + y = 10, for y:
y = 10 - 2x
Now we can substitute this value of y into the second equation, x – 3y = -2:
x – 3(10 - 2x) = -2
Simplify the equation:
x – 30 + 6x = -2
7x – 30 = -2
7x = 28
x = 4
Now we can substitute x = 4 back into the first equation to solve for y:
2(4) + y = 10
8 + y = 10
y = 2
The solution to the system is x = 4 and y = 2. This means the system has one unique solution.
Sure! I'd be happy to explain how to solve these systems of equations.
1. To verify if (-4, 12) is the solution to the system, we substitute the values of x and y into both equations and check if the equations are satisfied.
Given system:
2x + y = 5 ...(Equation 1)
-5x - 2y = -6 ...(Equation 2)
Substituting x = -4 and y = 12 into Equation 1:
2(-4) + 12 = 5
-8 + 12 = 5
4 = 5 (NOT true)
Substituting x = -4 and y = 12 into Equation 2:
-5(-4) - 2(12) = -6
20 - 24 = -6
-4 = -6 (NOT true)
Since both equations are not satisfied, (-4, 12) is not the solution to the system.
2. To solve the system by graphing, we plot both equations on a graph and find the point where they intersect. This point is the solution to the system.
Given system:
x + y = 2 ...(Equation 1)
2y - x = 10 ...(Equation 2)
To graph Equation 1, we rearrange it to solve for y:
y = 2 - x
To graph Equation 2, we rearrange it to solve for y:
y = (x + 10) / 2
Now we plot the graphs of both equations on the same coordinate system. The point where the graphs intersect is the solution to the system.
The solution to this system by graphing is (-4, 6).
3. To solve the system by substitution, we solve one equation for one variable and substitute that expression into the other equation. We then solve for the remaining variable.
Given system:
2x + y = 10 ...(Equation 1)
x - 3y = -2 ...(Equation 2)
Let's solve Equation 2 for x:
x = 3y - 2
Now substitute this expression for x in Equation 1:
2(3y - 2) + y = 10
6y - 4 + y = 10
7y - 4 = 10
7y = 14
y = 2
Substitute the value of y back into Equation 2:
x - 3(2) = -2
x - 6 = -2
x = 4
Thus, the solution to the system by substitution is (x, y) = (4, 2).
In summary:
- (-4, 12) is not the solution to the system.
- The solution to the system by graphing is (-4, 6).
- The solution to the system by substitution is (4, 2).