Problem 31 - Algebra 2

Question

a. Form a pair of simultaneous equations by letting \(y_{1}\) equal the left side and \(y_{2}\) equal the right side of \(\sqrt{5}-x=1\) . Graph the equations. b. Repeat part (a) with the equivalent equation \(\sqrt{5}=x+1\) c. Repeat part (a) with the equivalent equation \(\sqrt{5}-x-1=0\) d. Writing Describe the similarities and differences among the graphs of the three sets of simultaneous equations.

Step-by-Step Solution

Answer

The pairs of simultaneous equations for parts (a), (b), and (c) are \(y_1 = \sqrt{5}-x, y_2 = 1\), \(y_1 = \sqrt{5}-x, y_2 = 1-x\), and \(y_1 = \sqrt{5}-x-1, y_2 = 0\) respectively. After graphing, differences and similarities among the sets of equations are observed and described.

Step 1: Form the Pair of Simultaneous Equations for Part a

Start from the equation \(\sqrt{5}-x=1\). Let \(y_{1}=\sqrt{5}-x\) and \(y_{2}=1\). These are the pair of simultaneous equations.

Step 2: Graph the Equations for Part a

Graph the equations \(y_{1}=\sqrt{5}-x\) and \(y_{2}=1\) on a Cartesian plane. The points of intersection give the solution to the equations.

Step 3: Form the Pair of Simultaneous Equations for Part b

Starting from the equivalent equation \(\sqrt{5}=x+1\), let \(y_{1}=\sqrt{5}-x\) and \(y_{2}=1-x\). These form another pair of simultaneous equations.

Step 4: Graph the Equations for Part b

Now, graph these simultaneous equations \(y_{1}=\sqrt{5}-x\) and \(y_{2}=1-x\) on a Cartesian plane. Again, the points of intersection give the solutions to the equations.

Step 5: Form the Pair of Simultaneous Equations for Part c

Starting from the equivalent equation \(\sqrt{5}-x-1=0\), let \(y_{1}=\sqrt{5}-x-1\) and \(y_{2}=0\). These are the final pair of simultaneous equations.

Step 6: Graph the Equations for Part c

Plot the equations \(y_{1}=\sqrt{5}-x-1\) and \(y_{2}=0\) on a Cartesian plane. The points of intersection indicate the solutions to the equations.

Step 7: Comparison of the Graphs

Observe the three sets of graphs for any patterns or trends. Describe the similarities and differences among the graphs. First, notice whether the graphs intersect at the same or different points. Then compare the graph orientation and overall shape. Finally, mention if any trend was consistent across all three sets of equations.

Key Concepts

Graphing Equations

Graphing equations is like drawing a picture of a math problem. When you have an equation, you can create a visual by plotting it on a graph. This helps you to see the relationship between variables. In the context of simultaneous equations, graphing helps find where two equations meet, which is their common solution.
To graph an equation, follow these simple steps:

• First, rewrite the equation in a form where you can easily see what values you need to plot. This might mean solving for one variable.
• Next, choose some values for one of your variables and calculate the corresponding values of the other variable.
• Plot these calculated points on a coordinate plane.
• Finally, draw a line or curve through the points.

The points where two equations intersect on the graph show the solution to the simultaneous equations. It's a very visual way to solve equations!

Equation Solving

Solving equations involves finding the value of the variable that makes the equation true. In simultaneous equations, you have two different equations and you're looking for the one solution that fits both.
There are several methods to solve these, including graphing, substitution, and elimination. Here’s a brief overview:

• Graphing Method: As discussed earlier, this involves plotting each equation on the same graph and finding their intersecting point.
• Substitution Method: Solve one equation for one variable and substitute this value into the other equation. This reduces the system to one equation with one variable.
• Elimination Method: Manipulate the equations to eliminate one variable, leaving a solitary equation to solve for one variable.

Each of these methods has its own pros and cons but understanding each one will provide a comprehensive toolkit for solving simultaneous equations.

Coordinate Plane

The coordinate plane is a flat surface defined by two axes that intersect at the origin. It forms a grid where each point is represented by a pair of numbers, known as coordinates.
Here are the main features of the coordinate plane:

• The horizontal axis is called the x-axis.
• The vertical axis is called the y-axis.
• The point where the x-axis and y-axis meet is known as the origin, with coordinates (0,0).

When you graph an equation on a coordinate plane, each solution to the equation is represented by a point on the plane. The coordinate plane provides a visual method to solve equations and explore mathematical relationships through graphing.

Mathematical Analysis

Mathematical analysis involves breaking down mathematical expressions and functions to understand their properties. In the context of the exercise, this means interpreting the graphs of simultaneous equations and understanding their intersections.
Through mathematical analysis, you can:

• Determine if the graphs of equations are parallel or intersecting.
• Analyze the slope and position of lines to understand their relationships.
• Compare different forms of equivalent equations.

For example, in the original solution, three equivalent equations were analyzed. Each of these equations graphed differently, but mathematical analysis showed their similarities and the consistent solution point. This type of analysis is essential for identifying patterns in mathematical problems.