Solve the system of equations. 3x+y=2 7x−4y=30 (1 point)
To solve the system of equations, we can use either the substitution method or the elimination method.
Let's solve it using the elimination method:
First, we will multiply the first equation by 4 to make the coefficients of y in both equations equal:
4(3x + y) = 4(2)
12x + 4y = 8
The system of equations now becomes:
12x + 4y = 8
7x - 4y = 30
Next, we will add the two equations together to eliminate the y variable:
(12x + 4y) + (7x - 4y) = 8 + 30
12x + 7x = 38
19x = 38
x = 38/19
x = 2
Now, we substitute the value of x = 2 into one of the original equations, let's use the first equation:
3x + y = 2
3(2) + y = 2
6 + y = 2
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 (1 point)
To solve the system of equations, we can use either the substitution method or the elimination method.
Let's solve it using the elimination method:
First, we can multiply the second equation by 4 to make the coefficients of x equal:
4(x - 5y) = 4(-20)
4x - 20y = -80
The system of equations now becomes:
-4x + 6y = 38
4x - 20y = -80
Next, we add the two equations together to eliminate the x variable:
(-4x + 6y) + (4x - 20y) = 38 + (-80)
-14y = -42
Dividing both sides of the equation by -14, we get:
y = -42 / -14
y = 3
Now we can substitute the value of y = 3 into one of the original equations, let's use the second equation:
x - 5(3) = -20
x - 15 = -20
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 (1 point)
To determine how many solutions the system of equations has, we can look at the slopes of the equations or check if the lines are coincident (overlap) or parallel.
We can start by rewriting the equations in slope-intercept form (y = mx + b).
The first equation:
-5x + y = -18
y = 5x - 18
The second equation:
10x - 2y = 36
-2y = -10x + 36
y = 5x - 18
Upon rewriting, we see that both equations have the same slope and y-intercept. This means the lines are coincident (they are exactly the same line). Coincident lines have an infinite number of solutions, as they overlap at every single point.
Therefore, the system of equations has an infinite number of solutions.
To solve the system of equations, we can use the method of substitution or elimination. Let's use the elimination method.
First, we'll multiply the first equation by 4 to eliminate the y term:
4(3x + y) = 4(2)
12x + 4y = 8
Now, we can add this equation to the second equation:
(12x + 4y) + (7x − 4y) = 8 + 30
12x + 7x + 4y − 4y = 38
19x = 38
Divide both sides of the equation by 19 to solve for x:
19x/19 = 38/19
x = 2
Substitute the value of x back into either of the original equations. Let's use the first equation:
3x + y = 2
3(2) + y = 2
6 + y = 2
Subtract 6 from both sides of the equation:
6 + y - 6 = 2 - 6
y = -4
Therefore, the solution to the system of equations is x = 2 and y = -4.
To solve this system of equations, we can use the method of substitution or elimination. Let's use the method of substitution to find the values of x and y.
Step 1: Solve one of the equations for one variable in terms of the other.
Let's solve the first equation for y:
3x + y = 2
y = 2 - 3x
Step 2: Substitute this value of y into the second equation.
Now, substitute the value of y into the second equation:
7x - 4(2 - 3x) = 30
Step 3: Simplify and solve for x.
Using the distributive property, simplify the equation:
7x - 8 + 12x = 30
Combine like terms:
19x - 8 = 30
Add 8 to both sides:
19x = 38
Divide both sides by 19:
x = 38/19
x = 2
Step 4: Substitute x = 2 into one of the original equations.
Let's substitute the value of x into the first equation:
3(2) + y = 2
6 + y = 2
Subtract 6 from both sides:
y = -4
So the solution to the system of equations is x = 2 and y = -4.