Solve the System of Equations y=x+16 x^2+y^2=128
The given problem is asking to find the values of x and y that simultaneously satisfy two algebraic equations. The first equation is a linear equation that defines a relationship where y is equal to x plus 16. The second equation is a quadratic equation, specifically a circle equation in standard form, which represents the sum of the squares of x and y equalling 128. The task is to determine the exact numbers for x and y that make both equations true when those x and y values are plugged into them. This system of equations is likely to have a geometric interpretation on a coordinate plane, where the solution(s) represent the point(s) of intersection between a line and a circle.
$y = x + 16$$x^{2} + y^{2} = 128$
Answer
Solution:
Step 1: Substitute $y$ with $x + 16$ in the second equation.
Step 1.1: In the equation $x^2 + y^2 = 128$, replace $y$ with $x + 16$.
$$x^2 + (x + 16)^2 = 128$$ $$y = x + 16$$
Step 1.2: Expand and simplify the equation.
Step 1.2.1: Expand the squared term $(x + 16)^2$.
Step 1.2.1.1: Express $(x + 16)^2$ as $(x + 16)(x + 16)$.
$$x^2 + (x + 16)(x + 16) = 128$$ $$y = x + 16$$
Step 1.2.1.1.2: Use the FOIL method to expand $(x + 16)(x + 16)$.
Step 1.2.1.1.2.1: Distribute $x$ across $(x + 16)$ and $16$ across $(x + 16)$.
$$x^2 + x(x + 16) + 16(x + 16) = 128$$ $$y = x + 16$$
Step 1.2.1.1.2.2: Distribute $x$ and $16$ within the parentheses.
$$x^2 + x \cdot x + x \cdot 16 + 16 \cdot x + 16 \cdot 16 = 128$$ $$y = x + 16$$
Step 1.2.1.1.3: Combine like terms and simplify.
Step 1.2.1.1.3.1: Multiply $x$ by $x$ and $16$ by $16$.
$$x^2 + x^2 + 16x + 16x + 256 = 128$$ $$y = x + 16$$
Step 1.2.1.2: Add $x^2$ to $x^2$ and $16x$ to $16x$.
$$2x^2 + 32x + 256 = 128$$ $$y = x + 16$$
Step 2: Solve for $x$ in the simplified equation.
Step 2.1: Subtract $128$ from both sides of $2x^2 + 32x + 256 = 128$.
$$2x^2 + 32x + 128 = 0$$ $$y = x + 16$$
Step 2.2: Factor the quadratic equation.
Step 2.2.1: Factor out a $2$ from the terms.
$$2(x^2 + 16x + 64) = 0$$ $$y = x + 16$$
Step 2.2.2: Recognize the perfect square trinomial and factor it.
$$2(x + 8)^2 = 0$$ $$y = x + 16$$
Step 2.3: Solve for $x$ by setting the squared term equal to zero.
$$x + 8 = 0$$ $$x = -8$$
Step 3: Substitute $x = -8$ into the first equation to find $y$.
$$y = (-8) + 16$$ $$y = 8$$
Step 4: Write the solution as an ordered pair.
$(-8, 8)$
Step 5: Present the solution in different forms.
Point Form: $(-8, 8)$ Equation Form: $x = -8, y = 8$
Step 6: End of the solution process.
Solution:"The solution to the system of equations is the point (-8, 8)."
Knowledge Notes:
Substitution Method: This involves replacing one variable in an equation with an expression involving the other variable.
FOIL Method: A technique for expanding the product of two binomials, which stands for First, Outer, Inner, Last.
Perfect Square Trinomial: A trinomial of the form $a^2 + 2ab + b^2$, which can be factored into $(a + b)^2$.
Factoring: The process of breaking down a composite number into its prime factors or an expression into simpler expressions that can be multiplied together to result in the original expression.
Quadratic Equations: Equations of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $a \neq 0$. They can often be solved by factoring, completing the square, or using the quadratic formula.
System of Equations: A set of two or more equations with the same variables. The solution to the system is the set of values that satisfy all equations simultaneously.
