Solve for the simultaneous equation
X^2 + y - 5=0
Y - 7x + 3= 0
To solve the simultaneous equations:
1) X^2 + y - 5 = 0
2) y - 7x + 3 = 0
We can solve this system of equations by substitution. First, solve equation (2) for y:
y = 7x - 3
Now, substitute this value of y into equation (1):
x^2 + (7x - 3) - 5 = 0
Rearrange this equation to simplify:
x^2 + 7x - 8 = 0
Now, we can solve this quadratic equation by factoring or using the quadratic formula. In this case, the quadratic equation factors as:
(x + 8)(x - 1) = 0
Setting each factor equal to zero:
x + 8 = 0 --> x = -8
x - 1 = 0 --> x = 1
So, we have two possible x-values, x = -8 and x = 1. Now, substitute these values into equation (2) to solve for the corresponding y-values:
For x = -8:
y = 7(-8) - 3 = -56 - 3 = -59
For x = 1:
y = 7(1) - 3 = 7 - 3 = 4
Therefore, the solutions to the simultaneous equations are x = -8, y = -59 and x = 1, y = 4.
To solve for the simultaneous equations, we can use the method of substitution:
Step 1: Solve one equation for one variable in terms of the other variable.
Let's solve the second equation for y:
y - 7x + 3 = 0
y = 7x - 3
Step 2: Substitute the expression for y into the other equation.
Substitute y = 7x - 3 into the first equation:
x^2 + (7x - 3) - 5 = 0
Step 3: Simplify and solve for the remaining variable.
Rearrange the equation:
x^2 + 7x - 8 = 0
Step 4: Solve the quadratic equation.
Factor or use the quadratic formula to solve for x. In this case, let's use factoring:
(x - 1)(x + 8) = 0
Setting each factor equal to zero, we get:
x - 1 = 0 or x + 8 = 0
Solving for x, we find:
x = 1 or x = -8
Step 5: Substitute the values of x back into one of the original equations to solve for y.
Using y = 7x - 3:
For x = 1, y = 7(1) - 3 = 4
For x = -8, y = 7(-8) - 3 = -59
Therefore, the solutions for the simultaneous equations are:
(x, y) = (1, 4) and (-8, -59)