Problem 29 - Precalculus Mathematics 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 6. $$\left\\{\begin{array}{l}x+2 y=7 \\\5 x-y=2\end{array}\right.$$

Step-by-Step Solution

Answer

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

Step 1: Write Down the System of Equations

The given system of equations is: \( x + 2y = 7 \) and \( 5x - y = 2 \). We need to find if there is a unique solution, no solution, or infinitely many solutions.

Step 2: Solve the First Equation for x

From the first equation, we can express \( x \) in terms of \( y \): \( x = 7 - 2y \).

Step 3: Substitute x in the Second Equation

Substitute \( x = 7 - 2y \) into the second equation \( 5x - y = 2 \): \( 5(7 - 2y) - y = 2 \). Expand the terms: \( 35 - 10y - y = 2 \). Combine like terms: \( 35 - 11y = 2 \).

Step 4: Solve the Equation for y

Subtract 35 from both sides of the equation: \( -11y = 2 - 35 \) which simplifies to \( -11y = -33 \). Divide each side by -11 to solve for \( y \): \( y = 3 \).

Step 5: Substitute the Value of y to Find x

Now use the value of \( y \) in the expression \( x = 7 - 2y \). Substitute \( y = 3 \) into \( x = 7 - 2y \): \( x = 7 - 2(3) \). Simplify it to find \( x = 7 - 6 = 1 \).

Step 6: Write the Solution as an Ordered Pair

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

Key Concepts

Solving Linear Equations

Linear equations form the backbone of algebra. They are equations where the highest power of the variable is one. Here, we deal with a system of two linear equations: \( x + 2y = 7 \) and \( 5x - y = 2 \). To find the solution, we need to determine the values of \( x \) and \( y \) that make both equations true simultaneously.
This involves combining or manipulating these equations using algebraic techniques to isolate the variables.
The key is to transform these equations step by step until we solve for one variable, and then use that solution to find the other.
The goal is to find a point where both lines (represented by the equations) intersect, indicating a solution.

Unique Solution

A system of equations can have different types of solutions: a unique solution, no solution, or infinitely many solutions.
A unique solution means that there is exactly one pair \((x, y)\) that satisfies both equations.
In geometric terms, this means that the two lines represented by the equations intersect at a single point on the cartesian plane.

• If the system has a unique solution, it confirms that the lines are not parallel and cross exactly once.
• In our exercise, this single point of intersection is \((1, 3)\), meaning that substituting \(x = 1\) and \(y = 3\) into both equations satisfies them both.

This concept is crucial because it explains how two mathematically distinct lines can share exactly one common point.

Substitution Method

The substitution method is a fundamental algebraic tool used to solve systems of equations.
This approach involves expressing one variable in terms of another to facilitate substitution into another equation.
For our system, we first expressed \(x\) in terms of \(y\): \(x = 7 - 2y\).
Next, by substituting this expression for \(x\) into the second equation, \(5x - y = 2\), we could solve for \(y\).

• This method simplifies the process by reducing the number of variables in one equation, making it possible to solve for one variable completely.
• Once \(y\) was found, \(y = 3\), we substituted it back into the equation \(x = 7 - 2y\) to find the value of \(x\).

The substitution method is particularly useful when one of the equations can be easily isolated for one variable, providing a clear path to the answer.

Ordered Pairs

Ordered pairs are a vital concept when discussing solutions to systems of equations.
An ordered pair \((x, y)\) represents a point on a 2D coordinate plane, where \(x\) is the horizontal component and \(y\) is the vertical component.
In solving our system of equations, we found the ordered pair \((1, 3)\), which is the point where the two lines intersect and the unique solution to the system.

• Ordered pairs are used to denote specific locations on the plane and are essential in describing solutions that satisfy both equations in a system.
• They help us visualize problems graphically, providing a tangible representation of abstract algebraic solutions.
• Understanding and interpreting these pairs are crucial for graphing solutions and verifying the accuracy of your results.

An ordered pair solution means you've found the exact values for the variables that work in both original equations.