Problem 27 - 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. $$\left\\{\begin{aligned} -x+y &=2 \\ 4 x-3 y &=-3 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution is \((3, 5)\).

Step 1: Express One Variable in Terms of Another

From the first equation, \(-x + y = 2\), solve for \(y\). \(y = x + 2\).

Step 2: Substitute into the Second Equation

Substitute \(y = x + 2\) into the second equation, \(4x - 3(x + 2) = -3\).

Step 3: Simplify and Solve the Equation

Simplify the equation: \(4x - 3x - 6 = -3\) which becomes \(x - 6 = -3\). Solve for \(x\): \(x = 3\).

Step 4: Solve for the Other Variable

Use the value \(x = 3\) in the expression \(y = x + 2\): \(y = 3 + 2 = 5\).

Step 5: Write the Solution as an Ordered Pair

The solution to the system is \((x, y) = (3, 5)\).

Key Concepts

Substitution Method

The substitution method is a popular algebraic technique used to solve systems of linear equations. It involves solving one equation for a single variable and then substituting this expression into the other equation. This substitution allows you to work with just one variable, simplifying the process of finding the solution. Here is how the substitution method works:

• Firstly, pick one of the equations and express one variable in terms of the other. This step simplifies the equation, making it easier to manage.
• Secondly, substitute the expression obtained into the other equation. This changes the system to a single equation with one variable.
• After this, solve the resulting equation to find the value of one variable.
• Finally, use the value you found to substitute back in the expression you created initially to calculate the second variable.

By following these steps, you can find the values of the variables that satisfy both equations in the system, leading to the solution in a clear and methodical manner.

Linear Equations

Linear equations form the backbone of systems like the one we're working with. A linear equation is any equation that can be written in the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. Linear equations graph as straight lines on a coordinate plane.Key features of linear equations include:

• They have a consistent slope, which means they increase or decrease at a constant rate.
• The solutions are typically the points where these equations intersect, known as the solution set of a system.
• Examples of simple linear equations are \(y = 2x + 3\), where you can clearly see the slope and y-intercept.

When dealing with systems of linear equations, their solutions could be no solution, one solution, or infinitely many solutions. Understanding these characteristics helps in predicting and verifying the results when solving such systems.

Ordered Pairs

Ordered pairs are a fundamental concept in algebra, especially when representing solutions to systems of equations. An ordered pair is simply a pair of numbers, usually written in the form \((x, y)\), that represents the position of a point in the coordinate plane.Here’s what’s important to know about ordered pairs:

• The numbers in an ordered pair represent coordinates, where the first number is the x-coordinate (horizontal position) and the second is the y-coordinate (vertical position).
• In the context of solving systems of linear equations, ordered pairs signify the exact point where the lines intersect.
• Finding the ordered pair that solves a system means identifying the values of \(x\) and \(y\) that satisfy both equations at the same time.

Being comfortable with ordered pairs makes interpreting the results of a system of equations straightforward, helping to visualize and comprehend the solution visually and mathematically.