Problem 61 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the system graphically or algebraically. Explain your choice of method. $$\left\\{\begin{array}{l} 2 x-y=0 \\ x^{2}-y=-1 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 1\) and \(y = 2\).

Step 1: Re-arrange the linear equation

Rearrange the first equation, \(2x - y = 0\), to solve for \(y\). This gives \(y = 2x\).

Step 2: Substitute \(y\) into the quadratic equation

Substitute expression \(y = 2x\) from Step 1 into the quadratic equation \(x^{2} - y = -1\). This results in \(x^{2} - 2x = -1\).

Step 3: Re-arrange the quadratic equation to standard form

Re-arrange equation obtained in Step 2 to standard quadratic form. This gives us \(x^{2} - 2x + 1 = 0\). This can be expressed as \((x - 1)^{2} = 0\).

Step 4: Solve the quadratic equation

Solve \((x - 1)^{2} = 0\) for \(x\). This yields a single real root \(x = 1\).

Step 5: Substitute \(x\) back into the linear equation to find \(y\)

Substitute \(x = 1\) obtained from Step 4 into the rearranged linear equation from Step 1. This results in \(y = 2(1)\), therefore \(y = 2\).

Key Concepts

Graphical Method

The graphical method is a visual approach to solving systems of equations by plotting the equations on a graph. Each equation represents a curve. The intersection points of the curves correspond to the solutions of the system. When using this method:

• Convert each equation to a form suitable for graphing, usually by solving for y.
• Plot each equation on a coordinate plane.
• Look for points where the graphs intersect. These are your solutions.

For the given system:

• Rearrange the first equation, \(2x - y = 0\), to \(y = 2x\). This is a straight line.
• The second equation, \(x^2 - y = -1\), rearranges to \(y = x^2 + 1\). This is a parabola.
• Graph these on the same axes. The intersection point at \((1, 2)\) is where the line and parabola meet, representing the solution.

Graphical methods are especially helpful for visual learners or when dealing with systems that are easy to graph.

Algebraic Method

The algebraic method involves using mathematical operations to find the solutions to systems of equations without graphing. It's precise and preferred for solving complex systems.

• Start by rearranging equations or use elimination or substitution to simplify.
• Algebra helps when variables are easily isolated or when graphing is impractical.

For example, in our system:

• The equation \(2x - y = 0\) easily rearranges to \(y = 2x\).
• Substitute this into the quadratic equation to eliminate y and solve for x.

This method is efficient and gives exact solutions, as demonstrated by finding \(x = 1\) and \(y = 2\) in the example.

Substitution Method

The substitution method is a popular algebraic technique for solving systems of equations. It involves substituting an expression from one equation into another, thereby reducing the number of variables.Here's how you can apply the substitution method:

• Solve one of the equations for one variable. In this case, solve \(2x - y = 0\) for \(y\) to get \(y = 2x\).
• Substitute this expression for \(y\) into the second equation, \(x^2 - y = -1\).
• This substitution changes the equation to \(x^2 - 2x = -1\), simplifying it to one variable.
• Rearrange to standard quadratic form \(x^2 - 2x + 1 = 0\) and solve for \(x\).

Once \(x\) is found, substitute back to find \(y\). This method is effective for systems where one equation is easily solvable for one variable.

Quadratic Equation

Quadratic equations are polynomial equations of degree two. The standard form is \(ax^2 + bx + c = 0\). Solving quadratic equations can be done using:

• Factoring
• The quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
• Completing the square

In our problem, after substitution, we have \(x^2 - 2x + 1 = 0\). Since it factors to \((x - 1)^2 = 0\), it's straightforward. The solution is \(x = 1\), as it occurs twice in the factorized form. In general:

• Always check if the equation can be factored easily.
• Use the quadratic formula for more complex scenarios.

Understanding how to solve quadratic equations is crucial as they often appear in various mathematical contexts.