Problem 6 - Elementary Algebra

Question

Determine Whether an Ordered Pair is a Solution of a System of Equations. In the following exercises, determine if the following points are solutions to the given system of equations. $$ \left\\{\begin{array}{l} x+y=1 \\ y=\frac{2}{5} x \end{array}\right. $$ (a) \(\left(\frac{5}{7}, \frac{2}{7}\right)\) (b) (5,2)

Step-by-Step Solution

Answer

The point \( \left( \frac{5}{7}, \frac{2}{7} \right) \) is a solution; the point (5, 2) is not.

Step 1: Understand the system of equations

The given system of equations consists of two equations: 1) The sum of x and y is equal to 1: \[x + y = 1\]2) y is equal to \( \frac{2}{5} \, x \): \[y = \frac{2}{5}x\]

Step 2: Test the first point \( \left( \frac{5}{7}, \frac{2}{7} \right) \) in equation 1

Substitute \( x = \frac{5}{7} \) and \( y = \frac{2}{7} \) into the first equation: \[\frac{5}{7} + \frac{2}{7} = 1\]Calculate the left side:\[\frac{5}{7} + \frac{2}{7} = \frac{7}{7} = 1\]Since the equation holds true, the point satisfies the first equation.

Step 3: Test the first point \( \left( \frac{5}{7}, \frac{2}{7} \right) \) in equation 2

Substitute \( x = \frac{5}{7} \) and \( y = \frac{2}{7} \) into the second equation: \[y = \frac{2}{5}x\]Calculate the right side:\[\frac{2}{7} = \frac{2}{5} \left( \frac{5}{7} \right)\]\[\frac{2}{7} = \frac{2 \cdot 5}{5 \cdot 7} = \frac{2}{7}\]Since the equation holds true, the point satisfies the second equation.

Step 4: Conclusion for the first point

Since the point \( \left( \frac{5}{7}, \frac{2}{7} \right) \) satisfies both equations, it is a solution to the system of equations.

Step 5: Test the second point (5, 2) in equation 1

Substitute \( x = 5 \) and \( y = 2 \) into the first equation: \[5 + 2 = 1\]Calculate the left side:\[7 eq 1\]Since the equation does not hold true, the point does not satisfy the first equation.

Step 6: Conclusion for the second point

Since the point (5, 2) does not satisfy the first equation, it is not a solution to the system of equations.

Key Concepts

ordered pairs

An ordered pair, like \( (x, y) \), represents a point in the two-dimensional Cartesian coordinate system. This system uses two numbers to show the position of points. The first number is the x-coordinate (horizontal axis), and the second number is the y-coordinate (vertical axis).
For example:

• The ordered pair \( (\frac{5}{7}, \frac{2}{7}) \) means x = \frac{5}{7} and y = \frac{2}{7}.
• The ordered pair \( (5, 2) \) means x = 5 and y = 2.

Ordered pairs allow us to easily locate points on a graph and are essential to solving and understanding systems of equations.

solution verification

Solution verification is the process of checking if an ordered pair is a solution to a given system of equations. To verify, you substitute the values of x and y from the ordered pair into both equations and check if both equations are true.
Here’s how:

• Substitute x and y into the first equation, and calculate.
• See if the outcome satisfies the equation.
• Repeat the process for the second equation.

If both equations hold true after substituting the values from the ordered pair, then that ordered pair is a solution to the system of equations. If even one equation does not hold true, then the ordered pair is not a solution.
For example, with the ordered pair \( \left( \frac{5}{7}, \frac{2}{7} \right) \) in the given system:

• First equation: \( \frac{5}{7} + \frac{2}{7} = 1 \) is true.
• Second equation: \( \frac{2}{7} = \frac{2}{5} \left( \frac{5}{7} \right) \) is also true.

Therefore, \( \left( \frac{5}{7}, \frac{2}{7} \right) \) is a solution. But for the pair (5, 2):

• First equation: \( 5 + 2 eq 1 \)

This means (5, 2) is not a solution.

simultaneous equations

Simultaneous equations are a set of equations with multiple variables that you solve together. The solution to the simultaneous equations is an ordered pair that satisfies all equations within the system.
Consider the system:
\[ \begin\bs\text{{array}}{{l}} x + y = 1 \ y = \frac{2}{5} x \ end\bs\text{{array}} \]
Here’s a breakdown:

• The first equation says that when you add x and y, the result is 1.
• The second equation says that y is \frac{2}{5} of x.

To solve these simultaneously means finding values for x and y that make both equations true at the same time.
For instance, testing \left( \frac{5}{7}, \frac{2}{7} \right) we see:

• In equation 1, substituting these values gives:\br\ \frac{5}{7} + \frac{2}{7} = 1 \br\ Since this is true, it satisfies equation 1.
• In equation 2, substituting these values gives:\br\ \frac{2}{7} = \frac{2}{5} * \frac{5}{7} \br\ This is also true, thus satisfying equation 2.

Therefore, \( \left( \frac{5}{7}, \frac{2}{7} \right) \) is a true solution to the system of simultaneous equations.