Math Problem Statement
Solve these simultaneous equations to find 2 pairs of coordinates. 5-2x=y x^2+7x+25=y
Solution
We are given the following two equations:
Step 1: Set the two equations equal to each other
Since both expressions are equal to , we can set the right-hand sides equal to each other:
Step 2: Rearrange the equation
Move all terms to one side:
Simplify the equation:
Multiply through by to make the coefficient of positive:
Step 3: Solve the quadratic equation
Now, solve the quadratic equation using factoring, the quadratic formula, or completing the square. Here, we will factor:
Thus, the solutions for are:
Step 4: Find the corresponding -values
Now, substitute and into either of the original equations. Let's use .
For :
So, the first pair of coordinates is .
For :
So, the second pair of coordinates is .
Final Answer:
The two pairs of coordinates are:
Would you like further details on solving quadratic equations, or have any questions on this problem?
Here are some related questions to explore:
- How do you solve quadratic equations using the quadratic formula?
- What is the process of factoring a quadratic equation?
- How do you complete the square to solve quadratics?
- Can you graph the two equations and find where they intersect?
- How do you check the solutions of a quadratic equation?
Tip: When solving quadratic equations, always look for factoring first, as it's usually the quickest method!
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Math Problem Analysis
Mathematical Concepts
Algebra
Simultaneous Equations
Quadratic Equations
Formulas
y = 5 - 2x
y = x^2 + 7x + 25
Quadratic equation formula: ax^2 + bx + c = 0
Theorems
Quadratic Formula
Factoring Quadratics
Suitable Grade Level
Grades 9-11