is each ordered pair a solution of the given system? write yes or no.
1. y=6x+12
2x-y=4
(-4, -12)
2. Y= -3x
x=4y+1/2
(-1/2,3/2)
3.x+2y=2
2x+5y=2
(6,-2)
4. Solve the system by graphing. Check your solution.
x + y=3
x - y=-1
In order for an ordered pair to be a solution to a system of equations, it must satisfy all equations in the system. So, when you plug the x and the y into each equation, they BOTH must be true to be a solution. If one is true and the other is false, its NOT a solution.
For example, your 3rd question . .
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Is (6,-2) a solution?
x+2y=2
6+2(-2)=2?
6-4=2?
2=2 (TRUE!)
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2x+5y=2
2(6)+5(-2)=2?
12-10=2?
2=2 (TRUE!)
Therefore, the point (6,-2) is a solution
For Graphing, get both equations into y=mx+b. Plot the y-ntercept and then use your slope to find your second point. Draw your graph. Repeat for the other equation. See where the two graphs cross(intersect). That ordered pair is your solution!
To determine whether each ordered pair is a solution of the given system, we can substitute the values of x and y into each equation and evaluate if the equation holds true.
1. y = 6x + 12
2x - y = 4
Substitute x = -4 and y = -12 into the equations:
For the first equation:
-12 = 6(-4) + 12
-12 = -24 + 12
-12 = -12
The first equation holds true.
For the second equation:
2(-4) - (-12) = 4
-8 + 12 = 4
4 = 4
The second equation also holds true.
Both equations hold true for the ordered pair (-4, -12). Therefore, the answer is "Yes".
2. y = -3x
x = 4y + 1/2
Substitute x = -1/2 and y = 3/2 into the equations:
For the first equation:
3/2 = -3(-1/2)
3/2 = 3/2
The first equation holds true.
For the second equation:
-1/2 = 4(3/2) + 1/2
-1/2 = 6 + 1/2
-1/2 = 7/2
The second equation does not hold true.
One of the equations does not hold true for the ordered pair (-1/2, 3/2). Therefore, the answer is "No".
3. x + 2y = 2
2x + 5y = 2
Substitute x = 6 and y = -2 into the equations:
For the first equation:
6 + 2(-2) = 2
6 - 4 = 2
2 = 2
The first equation holds true.
For the second equation:
2(6) + 5(-2) = 2
12 - 10 = 2
2 = 2
The second equation holds true.
Both equations hold true for the ordered pair (6, -2). Therefore, the answer is "Yes".
4. Solve the system by graphing. Check your solution.
x + y = 3
x - y = -1
By graphing the lines representing the equations, we can find their point of intersection, which is the solution to the system.
Graphically, the lines intersect at the point (1, 2). Checking the solution, we substitute x = 1 and y = 2 into each equation:
For the first equation:
1 + 2 = 3
3 = 3
The first equation holds true.
For the second equation:
1 - 2 = -1
-1 = -1
The second equation holds true.
Both equations hold true for the ordered pair (1, 2). Therefore, the answer is "Yes".
To determine if each ordered pair is a solution of the given system, we substitute the values of x and y into the equations and check if the equations hold true. Let's go through each question:
1. Given system:
Equation 1: y = 6x + 12
Equation 2: 2x - y = 4
Substituting (-4, -12) into the equations:
Equation 1: -12 = 6(-4) + 12
-12 = -24 + 12
-12 = -12 (True)
Equation 2: 2(-4) - (-12) = 4
-8 + 12 = 4
4 = 4 (True)
Both equations hold true when substituting (-4, -12) into the system, so the answer is YES.
2. Given system:
Equation 1: y = -3x
Equation 2: x = 4y + 1/2
Substituting (-1/2, 3/2) into the equations:
Equation 1: 3/2 = -3(-1/2)
3/2 = 3/2 (True)
Equation 2: -1/2 = 4(3/2) + 1/2
-1/2 = 6 + 1/2
-1/2 = 12/2 + 1/2
-1/2 = 13/2 (False)
Equation 2 is not true when substituting (-1/2, 3/2) into the system, so the answer is NO.
3. Given system:
Equation 1: x + 2y = 2
Equation 2: 2x + 5y = 2
Substituting (6, -2) into the equations:
Equation 1: 6 + 2(-2) = 2
6 - 4 = 2
2 = 2 (True)
Equation 2: 2(6) + 5(-2) = 2
12 - 10 = 2
2 = 2 (True)
Both equations hold true when substituting (6, -2) into the system, so the answer is YES.
4. Solving the system by graphing:
Equation 1: x + y = 3
Equation 2: x - y = -1
To graph these equations, we can rearrange them into slope-intercept form (y = mx + b) and plot the lines:
Equation 1: y = -x + 3
Equation 2: y = x + 1
We can see that the lines intersect at the point (x, y) = (2, 1).
To check the solution, we substitute (2, 1) into the equations:
Equation 1: 2 + 1 = 3 (True)
Equation 2: 2 - 1 = -1 (True)
Both equations hold true when substituting (2, 1) into the system, confirming that it is a solution.
Therefore, the answers are:
1. Yes
2. No
3. Yes
4. Yes