Problem 79 - Thinking Mathematically

Question

Write a system of equations having \(\\{(-2,7)\\}\) as a solution set. (More than one system is possible.)

Step-by-Step Solution

Answer

Two possible equations forming a system with solution (-2,7) could be: y = x + 9 and y = -2x + 3.

Step 1: Create First Equation

An equation for the solution set \((-2, 7)\) will satisfy the equation when x is -2 and y is 7. A simple equation can be created with these values. Let's form the first equation: \[y = x + 9\] When x is -2, y will be 7, therefore this equation is valid

Step 2: Create Second Equation

Similarly, a second equation is created to satisfy x = -2 and y = 7. The equation can be chosen as: \[y = -2x + 3\] If we substitute x = -2 and y = 7 in the equation, we can validate that it holds.

Key Concepts

Solving Linear Systems

Linear systems consist of two or more linear equations that have common solutions. A solution to a linear system is an ordered pair, or a set of values, that makes each equation in the system true simultaneously. To solve linear systems, we can use different methods such as graphing, substitution, elimination, and matrix approaches.

When graphing, each equation represents a line, and the point where the lines intersect is the solution. This visual method is straightforward but may not be precise if the intersection doesn’t occur at exact grid coordinates. With algebraic approaches, such as the substitution method, we use algebraic manipulations to find the exact solution, eschewing potential inaccuracy from graphing.

Substitution Method

The substitution method is a technique used to solve systems of linear equations. It involves rearranging one equation to express one variable in terms of the other, and then substituting that expression into the other equation. This reduces the system to a single equation with one variable.

For example, if you have the system \(\begin{array}{l} y = x + 9 \ y = -2x + 3 \ \end{array}\), you can directly use the first equation’s expression for \(y\) in the second equation. This eliminates \(y\), making it possible to solve for \(x\), and then \(y\) can be found by substituting \(x\)'s value back into either of the original equations. This method is especially useful when one of the equations is already solved for one of the variables.

Algebraic Equations

Algebraic equations are mathematical statements showing the equality of two expressions. They contain variables, constants, and arithmetic operations. The simplest form is a linear equation in one variable, shown as \(ax + b = 0\), where \(a\) and \(b\) are constants, and \(x\) is the variable.

Algebraic equations can represent real-world scenarios, like the relationship between distance, speed, and time, or the balance of incomes and expenses in a budget. Solving these equations involves finding the value(s) of the variable(s) that make the equation true. Mastery of algebraic equations is fundamental for exploring more complex areas of mathematics and its applications in science and engineering.