Math Problem Statement

Find all solutions of the system of equations algebraically. Write your solutions as coordinate points. 1: y + 66 = 2x^2 +2x 16 = 2x - y

2: y = x(x - 3) y = 4

Solution

Let's solve each system of equations algebraically and write the solutions as coordinate points.

System 1:

y + 66 &= 2x^2 + 2x \quad \text{(Equation 1)} \\ 16 &= 2x - y \quad \text{(Equation 2)} \end{align*}$$ #### Step 1: Solve for $$y$$ in Equation 2. From Equation 2, solve for $$y$$: $$y = 2x - 16$$ #### Step 2: Substitute into Equation 1. Substitute $$y = 2x - 16$$ into Equation 1: $$(2x - 16) + 66 = 2x^2 + 2x$$ Simplify: $$2x + 50 = 2x^2 + 2x$$ Cancel out the $$2x$$ terms: $$50 = 2x^2$$ Divide by 2: $$x^2 = 25$$ Take the square root of both sides: $$x = 5 \quad \text{or} \quad x = -5$$ #### Step 3: Find corresponding $$y$$-values. For $$x = 5$$: $$y = 2(5) - 16 = 10 - 16 = -6$$ For $$x = -5$$: $$y = 2(-5) - 16 = -10 - 16 = -26$$ #### Solution to System 1: The solutions are: $$(5, -6) \quad \text{and} \quad (-5, -26)$$ --- ### System 2: $$\begin{align*} y &= x(x - 3) \quad \text{(Equation 1)} \\ y &= 4 \quad \text{(Equation 2)} \end{align*}$$ #### Step 1: Substitute $$y = 4$$ into Equation 1. $$4 = x(x - 3)$$ Simplify: $$4 = x^2 - 3x$$ Rearrange to form a quadratic equation: $$x^2 - 3x - 4 = 0$$ #### Step 2: Solve the quadratic equation. Factor the quadratic: $$(x - 4)(x + 1) = 0$$ Thus, $$x = 4$$ or $$x = -1$$. #### Step 3: Find corresponding $$y$$-values. For both values of $$x$$, $$y = 4$$ (since Equation 2 tells us that $$y = 4$$). #### Solution to System 2: The solutions are: $$(4, 4) \quad \text{and} \quad (-1, 4)$$ --- ### Final Solutions: - **System 1:** $$(5, -6)$$ and $$(-5, -26)$$ - **System 2:** $$(4, 4)$$ and $$(-1, 4)$$ Would you like further details on any steps or have any questions? --- ### Related Questions: 1. How do you solve quadratic equations by factoring? 2. What are alternative methods to solve quadratic equations if factoring doesn't work? 3. How can you solve systems of equations using substitution? 4. How can graphing systems of equations help visualize solutions? 5. What are some methods for solving non-linear systems of equations? **Tip:** When solving systems, always check both equations with your solutions to ensure accuracy.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Systems of Equations
Substitution Method

Formulas

Quadratic equation: ax^2 + bx + c = 0
Factoring method for solving quadratics

Theorems

Zero Product Property
Basic substitution in systems of equations

Suitable Grade Level

Grades 8-10

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