Solve each system using substitution. Tell whether the system has one solution, infinitely solutions, or no solution.
1.)
2x + y = -12;-4x - 2y = 30
2.)
1/3y = 7/3x + 5/3;x - 3y = 5
3.)
Y = X - 7;3x - 3y = 21
1.
2x + y = -12
-4x - 2y = 30 ---> 2x + y = -15
Clearly two parallel lines. I see the same slope, but different y-intercepts,
So obviously they cannot intersect
2.
1/3y = 7/3x + 5/3 ---> y = 7x + 5
x - 3y = 5
I see different slopes, so they WILL intersect
sub y = 7x + 5 into the 2nd
carry on
3.
Y = X - 7
3x - 3y = 21 ----> x - y = 7 or y = x - 7
They are both the same equation,
any equation will intersect with its self in an infinite number of points.
1.) To solve the system using substitution, we can solve one equation for one of the variables, and then substitute that expression into the other equation.
From the first equation, we can solve for y:
2x + y = -12
y = -2x - 12
Now we substitute this expression for y in the second equation:
-4x - 2(-2x - 12) = 30
-4x + 4x + 24 = 30
24 = 30
The equation 24 = 30 is not true, which means there is no solution to the system of equations.
Therefore, the system has no solution.
2.) Let's solve the system using substitution.
From the first equation, we can rewrite it as:
y = 7x + 5
Now we substitute this expression for y in the second equation:
x - 3(7x + 5) = 5
x - 21x - 15 = 5
-20x - 15 = 5
-20x = 20
x = -1
Substitute the value of x back into the equation y = 7x + 5:
y = 7(-1) + 5
y = -7 + 5
y = -2
Therefore, the system has one solution, which is x = -1 and y = -2.
3.) We can solve the system using substitution.
From the first equation, we can rewrite it as:
y = x - 7
Now we substitute this expression for y in the second equation:
3x - 3(x - 7) = 21
3x - 3x + 21 = 21
21 = 21
The equation 21 = 21 is true, which means that any value of x will satisfy the equation. Therefore, the system has infinitely many solutions.
In summary:
1.) The system has no solution.
2.) The system has one solution: x = -1 and y = -2.
3.) The system has infinitely many solutions.
To solve each system using substitution, we will solve one equation for one variable and then substitute it into the other equation. Let's go through each system step by step.
1.)
First, we'll solve the first equation for one variable. Let's solve it for x:
2x + y = -12
2x = -12 - y
x = (-12 - y)/2
Now, substitute this value of x into the second equation:
-4x - 2y = 30
-4((-12 - y)/2) - 2y = 30
-2(12 + y) - 2y = 30
-24 - 2y - 2y = 30
-24 - 4y = 30
-4y = 54
y = 54/(-4)
y = -13.5
Now, substitute this value of y back into the first equation to find x:
2x + (-13.5) = -12
2x - 13.5 = -12
2x = -12 + 13.5
2x = 1.5
x = 1.5/2
x = 0.75
The solution to the system is x = 0.75 and y = -13.5. Therefore, it has one unique solution.
2.)
First, let's solve the first equation for y:
(1/3)y = (7/3)x + (5/3)
Multiply both sides by 3 to remove the fraction:
y = 7x + 5
Now, substitute this value of y into the second equation:
x - 3(7x + 5) = 5
x - 21x - 15 = 5
-20x = 20
x = 20/(-20)
x = -1
Now, substitute this value of x back into the first equation to find y:
y = 7(-1) + 5
y = -7 + 5
y = -2
The solution to the system is x = -1 and y = -2. Therefore, it has one unique solution.
3.)
The first equation is already solved for y:
y = x - 7
Now, substitute this value of y into the second equation:
3x - 3(x - 7) = 21
3x - 3x + 21 = 21
0x + 21 = 21
21 = 21
In this case, both sides of the equation are equal, which means the equation is true for any value of x. It means there are infinitely many solutions for this system.
To summarize, the first system has one unique solution, the second system has one unique solution, and the third system has infinitely many solutions.