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Find the intersection points of a circle and a line.
Published on October 26, 2024
Algebra
The system of equations is solved by substitution, expanding, simplifying, using the quadratic formula to find x values, and then substituting back into the equations to find the corresponding y values. The solutions are (3, 4) and (-4, -3).
Question
x² + y² = 25 y = x + 1
Answer
Let’s solve this system of equations step-by-step.
1. Substitution
Since we already have an equation solved for ‘y’ (y = x + 1), we can substitute this expression into the first equation:
2. Expanding and Simplifying
Now, let’s expand the equation and combine like terms:
- Expand
: - Substitute back into the equation:
- Combine like terms:
- Subtract 25 from both sides:
3. Solving the Quadratic Equation
We now have a quadratic equation in the form
-
In our equation,
, , and . -
Substitute the values:
-
Simplify:
-
This gives us two possible solutions for ‘x’:
4. Finding the Corresponding ‘y’ Values
Now that we have two ‘x’ values, we can substitute them back into either of the original equations to find the corresponding ‘y’ values. Let’s use the simpler equation,
- For
: - For
:
Solution
Therefore, the solutions to the system of equations are: