Problem 47 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find a system of linear equations that has the given solution. (There are many correct answers.) \((a, a+4, a),\) where \(a\) is a real number

Step-by-Step Solution

Answer

The system of linear equations that fits the given solution is: 1. \(x = a\)2. \(y = x+4\)3. \(z = x\)

Step 1: Select the first equation

A simple equation that can give the solution \(a\) when solved would be \(x=a\). This is because \(x\) will equal to any number that \(a\) happens to be.

Step 2: Select the second equation

An equation that would give the solution \(a+4\) when solved would be \(y=x+4\). If we substitute \(x=a\) from the first equation into this equation, we get \(y=a+4\). Hence, this equation will fit the required solution.

Step 3: Select the third equation

Finally, for the last solution \(a\) can be obtained, a third equation needs to be considered. One such equation that fits the solution would be \(z=x\). This is because upon substituting \(x=a\) from the first equation into this, we get \(z=a\), thus fitting into the required solution.

Key Concepts

Linear Equations

Linear equations are among the simplest types of equations you'll encounter in mathematics. They are expressions that model a straight line when graphed on a Cartesian plane. Each linear equation can be written in the form of \(ax + b = 0\), where \(a\) and \(b\) are constants. In broader systems, these equations can involve multiple variables. A typical system might involve equations such as \(x + y = 5\) and \(2x - y = 3\).

Linear equations are characterized by each variable being to the first power. So, if you find an equation like \(x^2 + y = 7\), this is not a linear equation because \(x\) is squared. They are used extensively in various subjects, such as physics and engineering, to represent real-world phenomena.

Solutions of Equations

The solution of an equation is a vital concept in understanding algebraic expressions. It refers to the value or set of values that satisfy an equation. For a simple linear equation like \(x + 2 = 5\), the solution can be found by isolating \(x\). Subtract 2 from each side to get \(x = 3\).

• Single variable equations have a straightforward solution process.
• Systems of equations, on the other hand, involve more complex steps.

For systems, like two or more linear equations dealing with multiple variables, the solution is a set of values that satisfy all the equations simultaneously. For instance, in the exercise above, the solution involves determining values for \(x\), \(y\), and \(z\) that satisfy the whole system.

Substitution Method

The substitution method is a powerful tool for solving systems of linear equations. It involves solving one equation for a variable and substituting this value back into the other equation(s). This method simplifies the overall system and helps isolate variables. For instance, given the equations \(x = a\) and \(y = x + 4\), you can substitute \(x\) in the second equation to find \(y\).

This technique is particularly useful when one equation is already solved for a variable, as seen in the step-by-step solution where \(x\) is set to equal \(a\). You can then effectively substitute \(x = a\) into other equations like \(z = x\) to find \(z = a\) as well. It’s a method that keeps calculations more straightforward, avoiding complex algebraic manipulations.

Real Numbers

Real numbers are all the numbers that can be found on the number line. This includes rational numbers, such as fractions and integers, and irrational numbers like \(\pi\) and the square root of non-perfect squares. Most of the equations you solve every day in practice use real numbers.

• They form the basis of real-world calculations and measurements.
• Includes whole numbers, decimals, and both positive and negative numbers.

In the exercise problem, the solution involves real numbers since \(a\) is defined as a real number. This means any feasible solution will incorporate an actual measured value. Understanding the concept of real numbers provides a foundation for comprehending more complex mathematical subjects.