Problem 15 - Precalculus: Mathematical for Calculus

Question

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 3. $$\left\\{\begin{aligned} x+2 y &=7 \\ 5 x-y &=2 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The system has a unique solution: \( (1, 3) \).

Step 1: Write the system of equations

Given the system of equations: \( x + 2y = 7 \) and \( 5x - y = 2 \).

Step 2: Solve for one variable

We begin by solving the first equation for \( x \): \( x = 7 - 2y \).

Step 3: Substitute into the second equation

Substitute \( x = 7 - 2y \) into the second equation \( 5x - y = 2 \), yielding: \( 5(7 - 2y) - y = 2 \).

Step 4: Simplify the equation

Expand and simplify: \( 35 - 10y - y = 2 \), becoming \( 35 - 11y = 2 \).

Step 5: Solve for \( y \)

Rearrange the equation to solve for \( y \): \( -11y = 2 - 35 \) gives \( -11y = -33 \). Divide both sides by -11 to find \( y = 3 \).

Step 6: Find \( x \) using the value of \( y \)

Substitute \( y = 3 \) into the expression for \( x \): \( x = 7 - 2(3) = 7 - 6 = 1 \).

Step 7: Write the solution as an ordered pair

The solution to the system of equations is \( (x, y) = (1, 3) \).

Key Concepts

Linear Equations

Linear equations are mathematical expressions that describe a straight line when plotted on a graph. They consist of constants, variables, and coefficients without exceeding the power of one for each variable.
For instance, in the given exercise where we have the system of equations: \( x + 2y = 7 \) and \( 5x - y = 2 \), each equation represents a line on a graph.

• The term 'linear' implies that each equation features the variable with an exponent of one.
• Every linear equation can be arranged in the form \( ax + by = c \), known as the standard form, where \( a \), \( b \), and \( c \) are constants.
• Linear equations can have one solution, infinitely many solutions, or no solution at all depending on how the lines intersect.

Understanding the properties of linear equations can help us find where two lines intersect, which is the heart of solving systems of equations. In this task, both equations combine to create a system that can be solved to find a unique pair of \( x \) and \( y \) values.

Substitution Method

The substitution method is a technique used to solve systems of equations, especially useful when dealing with linear equations. It involves replacing one variable with an expression involving the other variable.
It's generally employed by first solving one of the equations for one of the variables and then substituting that expression into the other equation.

• In this case, we start with the first equation, \( x + 2y = 7 \), and solve for \( x \) to get \( x = 7 - 2y \).
• Next, we substitute \( x = 7 - 2y \) into the second equation, \( 5x - y = 2 \), effectively reducing the system to a single equation with one variable \( y \).
• This allows us to solve for \( y \) more straightforwardly, yielding \( y = 3 \).

By substituting back the value of \( y \) into the expression for \( x \), we then find \( x = 1 \). This method ensures that both equations are satisfied simultaneously, providing us with a comprehensive solution.

Ordered Pair Solution

An ordered pair solution represents the values of \( x \) and \( y \) that solve both equations in a system and can be written in the form \((x, y)\). This points to their position on a graph where the two lines intersect.
Ordered pair solutions are essential in presenting the outcome of solving a system of equations.

• For our exercise, we've determined that the ordered pair \((1, 3)\) satisfies both \( x + 2y = 7 \) and \( 5x - y = 2 \).
• This means if you graph these two equations, their crossing point will have the coordinates \( (1, 3) \).
• Ordered pairs show the exact point where the solution is valid, representing an intersection of the two equations on the Cartesian plane.

By confirming the ordered pair, we verify the accuracy and validity of our solution, ensuring it complies with the conditions laid out by both linear equations in the system.