What are the solutions to the system? (Hint: graphing is the easiest!)
y = x2 + 2x - 45
y = -3x + 5
Next easiest is by comparison.
Solve the quadratic by factoring
-3x+5 = x² +2x -45
x²+5x-50=0
(x+10)(x-5)=0
x=5 or x=-10
Substitute x to find corresponding values of y.
thank you peeps
To find the solutions to the system of equations, the easiest method is to graph both equations and observe where they intersect. Here's how you can graph them:
1. Start by rearranging each equation in the form y = mx + b, where m represents the slope and b represents the y-intercept.
For the first equation, y = x^2 + 2x - 45:
- To find the slope, differentiate the equation with respect to x: dy/dx = 2x + 2.
- Set the derivative equal to zero and solve for x to find the vertex of the parabola: 2x + 2 = 0 ⇒ x = -1.
- Calculate the y-coordinate of the vertex by substituting the x-value into the original equation: y = (-1)^2 + 2(-1) - 45 = -44.
- The vertex form of the equation is y = (x - (-1))^2 - 44.
- From this form, we can deduce that the vertex is (-1, -44) and thus the y-intercept is -44.
For the second equation, y = -3x + 5:
- Here, the slope is already apparent: m = -3.
- The y-intercept is b = 5.
2. Plot the y-intercepts for both equations on a coordinate plane.
- For the first equation, plot the point (-1, -44).
- For the second equation, plot the point (0, 5) since it represents the y-intercept.
3. Determine the direction and shape of the lines.
- For the first equation, y = x^2 + 2x - 45, since the coefficient of x^2 is positive, the parabola opens upwards.
- For the second equation, y = -3x + 5, since the slope is negative, the line slopes downwards.
4. Draw the graphs of both equations on the same coordinate plane.
Once the graphs are plotted, look for their intersection point(s). These point(s) represent the solutions to the system of equations.
well, it is hard for me to graph here so
eliminate y by subtracting the second equation from the first
x^2 + 5 x -50 = 0
(x-5)(x+10) = 0
so
x = 5 or x = -10
if x = 5, then y = -10
if x = -10 then y = 35