Solve the following system of equations using the elimination method. 2x−3y=14 , 4x+3y=10
To solve the system of equations using the elimination method, we can add the two equations together. This will help us eliminate the y term.
Adding the two equations:
(2x - 3y) + (4x + 3y) = 14 + 10
6x = 24
To isolate x, we divide both sides of the equation by 6:
6x/6 = 24/6
x = 4
Now that we have 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:
2x - 3y = 14
2(4) - 3y = 14
8 - 3y = 14
Subtract 8 from both sides of the equation:
-3y = 14 - 8
-3y = 6
Divide both sides of the equation by -3:
-3y / -3 = 6 / -3
y = -2
So the solution to the system of equations is x = 4 and y = -2.
Use the elimination method to solve the linear-quadratic system y=x^2+2x−4 , y=−x−6 .
To solve the system of equations using the elimination method, we'll equate the expressions of y in both equations and solve for x.
Since both equations are already in the form of "y = ...", we can equate the right sides:
x^2 + 2x - 4 = -x - 6
Next, we'll rearrange the equation to bring all terms to one side:
x^2 + 2x + x - 4 + 6 = 0
We combine like terms:
x^2 + 3x + 2 = 0
Now we have a quadratic equation. To solve it, we'll factor or use the quadratic formula. In this case, the equation factors as:
(x + 1)(x + 2) = 0
This gives us two possible solutions:
x + 1 = 0 or x + 2 = 0
Solving these equations separately, we find:
For x + 1 = 0, subtracting 1 from both sides gives x = -1.
For x + 2 = 0, subtracting 2 from both sides gives x = -2.
So the solutions for x are x = -1 and x = -2.
To find the corresponding y-values, we'll substitute each value of x into one of the original equations. Let's use the second equation:
For x = -1:
y = -x - 6
y = -(-1) - 6
y = 1 - 6
y = -5
For x = -2:
y = -x - 6
y = -(-2) - 6
y = 2 - 6
y = -4
So the solutions for the system of equations are x = -1, y = -5 and x = -2, y = -4.
To solve the system of equations using the elimination method, you need to eliminate one of the variables by adding or subtracting the two equations.
In this case, let's eliminate the variable "y". To do this, you can add the two equations together.
(2x - 3y) + (4x + 3y) = 14 + 10
Combine like terms:
6x = 24
Next, you can solve for "x" by dividing both sides of the equation by 6:
6x/6 = 24/6
x = 4
Now that we have found the value of "x", we can substitute it back into one of the original equations to find the value of "y". Let's use the first equation:
2x - 3y = 14
2(4) - 3y = 14
8 - 3y = 14
Subtract 8 from both sides:
-3y = 6
Divide both sides by -3:
y = -2
Therefore, the solution to the system of equations is x = 4 and y = -2.