Problem 7 - College Algebra

Question

Solve the system of nonlinear equations using substitution. $$ \begin{aligned} y &=x-3 \\ x^{2}+y^{2} &=9 \end{aligned} $$

Step-by-Step Solution

Answer

The solutions are \((0, -3)\) and \((3, 0)\).

Step 1: Express y in terms of x

From the first equation, we have that \(y = x - 3\). This expression will allow us to substitute \(y\) with \(x - 3\) in the second equation.

Step 2: Substitute y in second equation

Substitute \(y = x - 3\) into the second equation, \(x^2 + y^2 = 9\). This gives us \(x^2 + (x - 3)^2 = 9\).

Step 3: Expand and simplify

Expand \((x - 3)^2\) which is \(x^2 - 6x + 9\). Substitute back to the equation: \(x^2 + x^2 - 6x + 9 = 9\). Simplify to get \(2x^2 - 6x + 9 = 9\).

Step 4: Solve the simplified equation

Subtract 9 from both sides to get \(2x^2 - 6x = 0\). Factor out \(2x\) to get \(2x(x - 3) = 0\). This gives two solutions: \(x = 0\) and \(x = 3\).

Step 5: Solve for y using each x

For \(x = 0\), substitute into \(y = x - 3\) to get \(y = -3\). For \(x = 3\), substitute into \(y = x - 3\) to get \(y = 0\).

Step 6: Verify the solutions

Check each pair \((x, y)\) against the original equations. For \((0, -3)\), substitute into \(x^2 + y^2 = 9\): \(0^2 + (-3)^2 = 9\) holds true. For \((3, 0)\), substitute into the same equation: \(3^2 + 0^2 = 9\) holds true.

Key Concepts

Substitution Method

The substitution method is a powerful technique utilized to solve systems of equations. Especially when dealing with pairs of equations, it can simplify the problem significantly by reducing the number of variables. Here, we have a system composed of a linear equation and a nonlinear equation. This method is particularly beneficial since it allows us to express one variable in terms of another, which reduces the complexity of equations:

• Start by isolating one variable in one of the equations. In this exercise, the first equation is linear and simple: \( y = x - 3 \).
• Substitute the expression for the isolated variable into the other equation. This will transform and simplify the nonlinear equation into a form that is easier to solve.
• Solve for the remaining variable, and once determined, find the counterpart variable using the isolated expression from the first step.

This process not only helps in finding solutions but also in verifying them by substituting back to check their correctness in the original system of equations.

Nonlinear Equations

Nonlinear equations are those in which the variables do not form a straight line when plotted on a graph. They can include quadratic equations or involve curves like circles, ellipses, etc. In our exercise, one equation is a circle described by:\[x^2 + y^2 = 9\]Here, it represents a circle with a radius of 3, centered at the origin. Nonlinear systems can present additional complexity because their solutions might not be as straightforward:

• They can have more than one solution or even no solutions at all.
• The solutions may occur at intersections of lines and curves, curves and curves, etc.

Using the substitution method helps by converting one of the nonlinear equations into a simpler quadratic, which can more easily be tackled through algebraic manipulation.

Algebraic Expressions

Understanding algebraic expressions is key to smoothly working through systems of equations. In our given problem, the equation \( x^2 + (x - 3)^2 = 9 \) is an expression composed of terms that include constants and variables:

• Each term constitutes a component of the expression and should be manipulated carefully during expansion and simplification.
• Terms like \((x - 3)^2\) must be expanded using the distributive property: \( (x - 3)^2 = x^2 - 6x + 9 \).
• Combine like terms for simplification to form a solvable equation, such as reducing to \(2x^2 - 6x = 0\).

Handling algebraic expressions requires care and attention to detail, ensuring that each step follows algebraic rules and accurately represents the values they describe. Correctly managing these calculations ultimately leads to finding and verifying viable solutions for the system of equations.