Problem 48 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find a system of linear equations that has the given solution. (There are many correct answers.) \((3 a, a, a+2),\) where \(a\) is a real number

Step-by-Step Solution

Answer

A possible system of linear equations that fulfills the given conditions is:\[\begin{align*}1) &\quad x = 3a \2) &\quad y = a \3) &\quad z = a + 2\end{align*}\]However, there are numerous other correct systems. All that matters is that each equation delivers the appropriate component of the solution when \(a\) is input.

Step 1: Formulate the first equation

Begin with the formulation of an equation that will yield \(3a\) when solved for a single variable. An example would be \(𝑥 = 3a\).

Step 2: Formulate the second equation

Next is to establish a second equation which will yield \(a\) when solved. One possibility is \(y = a\).

Step 3: Formulate the third equation

Finally, build an equation that will provide \(a+2\) when solved. An example is \(z = a + 2\).

Key Concepts

Linear Equations

When dealing with systems of linear equations, we're looking at equations that graph as straight lines. A linear equation is an equation of the first order. This means that each variable in the equation is raised to the power of one. The simplest form of a linear equation in two variables is written as \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. It's called "linear" because its graph is a straight line.

Linear equations are used to describe relationships between different quantities. For example, in the context of the given exercise, the equations \(x = 3a\), \(y = a\), and \(z = a + 2\) are all linear, as each equation describes a straight-line relationship between variables and the unknown \(a\).

• Each equation gives us a straight line when plotted.
• They can be solved for one or more variables.
• Used to find the intersection points of lines (solutions).

Solution Sets

A solution set is the collection of all possible solutions that satisfy a given system of equations. For a system of linear equations, the solution set can be geometrically interpreted as the point(s) where the graphs of the equations intersect. In our exercise, we determine the solution set for the system that includes these equations:

• \(x = 3a\)
• \(y = a\)
• \(z = a + 2\)

Given these, the solution set will depend on the values assigned to \(a\), since \(a\) can take any real number value.

This results in an infinite number of solutions! Any value for \(a\) that you choose will satisfy all three equations simultaneously. Therefore, the solution set is deeply dependent on the parameter \(a\), and can include endless possibilities based on its definition.

Variables in Equations

In the context of linear equations, variables are symbols used to represent unknown values. When we form equations, variables allow the relationships between terms to be explored and analyzed. Our system of linear equations uses variables \(x\), \(y\), and \(z\), in addition to the parameter \(a\).

Here’s how they work within the given system:

• \(x\) relates to \(a\) through the expression \(3a\).
• \(y\) is equivalent to \(a\), establishing a direct correspondence.
• \(z\) is tied to \(a\) via the formula \(a + 2\).

Variables are crucial, as they allow us to manipulate and solve equations to find unknown values. Understanding the role of each variable in a system helps determine how they interact with one another for consistent solutions.

Real Numbers

Real numbers include all numbers on the number line, encompassing both rational and irrational numbers. In mathematics, they constitute all the solutions to a system of linear equations. Here, \(a\) is described as a real number, which means it can take any value from this vast, continuous set.

Real numbers have several key properties:

• They are ordered, meaning any two real numbers can be compared.
• They follow the laws of arithmetic: addition, subtraction, multiplication, and division.
• They provide a complete number set, filling every point on the number line without gaps.

In our exercise, \(a\) is pivotal as its value directly influences \(x\), \(y\), and \(z\). Because \(a\) can be any real number, it demonstrates the flexibility and broad application of systems constructed in such a manner.