Solving the System of Equations
To solve the system of equations:
- Equation 1: y = x² + 3
- Equation 2: y = x + 5
we can use a substitution method since both equations equal y. Here are the detailed steps:
Step 1: Set the equations equal to each other
Since both equations are equal to y, we can set them equal to each other:
x² + 3 = x + 5
Step 2: Rearrange the equation
Now, we will rearrange the equation to bring all terms to one side:
x² – x + 3 – 5 = 0
This simplifies to:
x² – x – 2 = 0
Step 3: Factor the quadratic equation
Next, we will factor the quadratic equation:
(x – 2)(x + 1) = 0
Step 4: Solve for x
Now we can set each factor to zero and solve for x:
- x – 2 = 0 → x = 2
- x + 1 = 0 → x = -1
Step 5: Find the corresponding y values
Next, we will substitute these values of x back into either original equation to find the corresponding y values.
For x = 2:
y = 2² + 3 = 4 + 3 = 7
For x = -1:
y = (-1)² + 3 = 1 + 3 = 4
Step 6: Write the solution as ordered pairs
Thus, the solutions to the system of equations are:
- (2, 7)
- (-1, 4)
These ordered pairs represent the points at which both equations intersect. Hence, we conclude that:
Final Answer
The solutions to the system of equations y = x² + 3 and y = x + 5 are:
- (2, 7)
- (-1, 4)