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:

  1. 52x=y5 - 2x = y
  2. x2+7x+25=yx^2 + 7x + 25 = y

Step 1: Set the two equations equal to each other

Since both expressions are equal to yy, we can set the right-hand sides equal to each other:

52x=x2+7x+255 - 2x = x^2 + 7x + 25

Step 2: Rearrange the equation

Move all terms to one side:

52xx27x25=05 - 2x - x^2 - 7x - 25 = 0

Simplify the equation:

x29x20=0-x^2 - 9x - 20 = 0

Multiply through by 1-1 to make the coefficient of x2x^2 positive:

x2+9x+20=0x^2 + 9x + 20 = 0

Step 3: Solve the quadratic equation

Now, solve the quadratic equation x2+9x+20=0x^2 + 9x + 20 = 0 using factoring, the quadratic formula, or completing the square. Here, we will factor:

(x+4)(x+5)=0(x + 4)(x + 5) = 0

Thus, the solutions for xx are:

x=4orx=5x = -4 \quad \text{or} \quad x = -5

Step 4: Find the corresponding yy-values

Now, substitute x=4x = -4 and x=5x = -5 into either of the original equations. Let's use y=52xy = 5 - 2x.

For x=4x = -4:

y=52(4)=5+8=13y = 5 - 2(-4) = 5 + 8 = 13

So, the first pair of coordinates is (4,13)(-4, 13).

For x=5x = -5:

y=52(5)=5+10=15y = 5 - 2(-5) = 5 + 10 = 15

So, the second pair of coordinates is (5,15)(-5, 15).

Final Answer:

The two pairs of coordinates are:

  1. (4,13)(-4, 13)
  2. (5,15)(-5, 15)

Would you like further details on solving quadratic equations, or have any questions on this problem?

Here are some related questions to explore:

  1. How do you solve quadratic equations using the quadratic formula?
  2. What is the process of factoring a quadratic equation?
  3. How do you complete the square to solve quadratics?
  4. Can you graph the two equations and find where they intersect?
  5. 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