solve the two equations using substitution.
y = 2+ 3x
4x - 3y = -16
First, we solve the first equation for y:
y = 2 + 3x
Next, we substitute this value of y into the second equation:
4x - 3(2 + 3x) = -16
Simplifying the equation:
4x - 6 - 9x = -16
Combining like terms:
-5x - 6 = -16
Add 6 to both sides:
-5x = -10
Finally, divide both sides by -5 to solve for x:
x = 2
Now, substitute this value of x back into the first equation to solve for y:
y = 2 + 3(2)
y = 2 + 6
y = 8
Therefore, the solution to the system of equations is x = 2 and y = 8.
solve the system of equations
2x + 6y = -18
x = 5y - 1
To solve this system of equations using substitution, we start by solving the second equation for x:
x = 5y - 1
Next, we substitute this expression for x into the first equation:
2(5y - 1) + 6y = -18
Distribute the 2:
10y - 2 + 6y = -18
Combine like terms:
16y - 2 = -18
Add 2 to both sides:
16y = -16
Divide both sides by 16:
y = -1
Now, substitute the value of y back into the second equation to solve for x:
x = 5(-1) - 1
x = -5 - 1
x = -6
Therefore, the solution to the system of equations is x = -6 and y = -1.
solve the system of equations
3x + y = 2
7x - 4y = 30
To solve this system of equations, we can use the method of substitution or elimination. Here, we will use the substitution method:
Let's solve equation 1 for y:
y = 2 - 3x
We substitute this expression for y into equation 2:
7x - 4(2 - 3x) = 30
Distribute the negative 4:
7x - 8 + 12x = 30
Combine like terms:
19x - 8 = 30
Add 8 to both sides:
19x = 38
Divide both sides by 19:
x = 2
Now substitute this value of x back into equation 1 to find y:
3(2) + y = 2
6 + y = 2
Subtract 6 from both sides:
y = 2 - 6
y = -4
Therefore, the solution to the system of equations is x = 2 and y = -4.
solve the system of equations
-4x + 6y = 38
x - 5y = -20
To solve this system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method:
First, we multiply the second equation by 4 to match the coefficient of x in the first equation:
4(x - 5y) = 4(-20)
4x - 20y = -80
Now we have the system of equations:
-4x + 6y = 38
4x - 20y = -80
Adding the two equations together eliminates the x term:
(-4x + 6y) + (4x - 20y) = 38 + (-80)
Simplifying:
-4x + 4x + 6y - 20y = 38 - 80
Combine like terms:
-14y = -42
Divide both sides by -14:
y = -42 / -14
y = 3
Now, substitute this value of y back into either of the original equations to solve for x:
x - 5(3) = -20
x - 15 = -20
Add 15 to both sides:
x = -20 + 15
x = -5
Therefore, the solution to the system of equations is x = -5 and y = 3.
how many solution(s) does the system of equations have?
-5x + y = -18
10x - 2y = 36
To determine how many solutions the system of equations has, we can analyze the slopes of the equations or check if the lines intersect.
For this system:
-5x + y = -18
10x - 2y = 36
Both equations can be simplified to the form y = mx + b by solving for y.
The first equation:
-5x + y = -18
y = 5x -18
The second equation:
10x - 2y = 36
-2y = -10x + 36
y = 5x - 18
Both equations simplify to y = 5x - 18, indicating that the two lines are identical. Therefore, the system has infinitely many solutions, as all points along the line satisfy both equations.
To solve the given system of equations using substitution, we'll start by solving one of the equations for one variable and then substituting that expression into the other equation. Let's solve the first equation (y = 2 + 3x) for y:
y = 2 + 3x
Now, let's substitute this expression for y in the second equation (4x - 3y = -16):
4x - 3(2 + 3x) = -16
Next, distribute the -3:
4x - 6 - 9x = -16
Combine like terms by subtracting:
-5x - 6 = -16
Now, isolate x by adding 6 to both sides:
-5x = -10
Divide both sides by -5:
x = 2
Now that we have found the value of x, we can substitute it back into one of the original equations to solve for y. Let's use the first equation (y = 2 + 3x):
y = 2 + 3(2)
Simplify by multiplying:
y = 2 + 6
y = 8
Thus, the solution to the system of equations is x = 2 and y = 8.