Problem 28 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the system by the method of substitution. Use a graphing utility to verify your results. $$\left\\{\begin{aligned} -\frac{2}{3} x+y &=2 \\ 3 x-\frac{1}{2} y &=4 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution for the system of equations is \(x = \frac{15}{8}\) and \(y = \frac{5}{4}\)

Step 1: Isolate one variable

In the first equation, isolate the variable 'y'. That turns \(-\frac{2}{3} x + y = 2\) into \(y = \frac{2}{3}x + 2\)

Step 2: Substitute in second equation

Substitute \(y\) from first equation into the second equation. So, \(3x - \frac{1}{2} (\frac{2}{3}x + 2) = 4\) turns into \(3x - \frac{1}{3}x -1 = 4\)

Step 3: Solve for x

Simplify and solve for 'x'. So, \(\frac{8}{3} x = 5\) yields \(x = \frac{15}{8}\)

Step 4: Substitute x to find y

Substitute \(x = \frac{15}{8}\) into the \(y\) equation to find 'y'. So, \(y = \frac{2}{3}\) * \(\frac{15}{8} + 2\) turns into \(y = \frac{5}{4}\).

Step 5: Verification with graphing utility

A graphing utility can be utilized to verify the obtained solution. Plotting the equations \(-2/3 x + y = 2\) and \(3x - 1/2 y = 4\), the intersection point should be the point \((15/8, 5/4)\).

Key Concepts

Method of Substitution

The method of substitution is a fundamental technique to solve systems of equations, particularly when dealing with two variables.
It involves a step-by-step process to replace one variable with an equivalent expression, allowing us to work with a single equation at a time. Let's break it down:

• First, choose one of the equations and isolate either of the variables, say 'y'. This creates a manageable expression where 'y' is expressed in terms of 'x'. For example, from \(-\frac{2}{3} x + y = 2\), we derive \(y = \frac{2}{3}x + 2\).
• Next, substitute the expression for 'y' into the other equation, turning it into a single-variable equation. This reduces \(3x - \frac{1}{2} y = 4\) to \(3x - \frac{1}{3}x -1 = 4\).
• Finally, solve this equation to find the value of 'x'. In this case, we find \(x = \frac{15}{8}\).

After finding 'x', substitute it back into the expression for 'y' to determine its value. This method is especially handy when one equation is much simpler than the other or when isolating a variable is straightforward.

Graphing Utility Verification

Once you have a solution from substitution or another algebraic method, verifying with a graphing utility ensures accuracy.
Graphing utilities, such as Desmos or a calculator capable of graphing, allow us to visualize equations as lines in the coordinate plane.Here's how verification works:

• Plot each equation on the same set of axes. They will appear as lines on the graph. In this exercise, we graph \(-\frac{2}{3} x + y = 2\) and \(3x - \frac{1}{2} y = 4\).
• Observe where the two lines intersect. The intersection point represents the solution \((x, y)\) that satisfies both equations simultaneously.
• Verify that the intersection point \(\left(\frac{15}{8}, \frac{5}{4}\right)\) matches the calculated solution.

This technique not only confirms your algebraic solution but also aids in visualizing the relationship between equations and solutions.

Linear Equations

Linear equations are equations of the first order that represent straight lines when graphed on a coordinate plane.
They are typically formatted as \(ax + by = c\), where ''a', 'b', and 'c' are constants.Key properties of linear equations include:

• A linear equation in two variables (like 'x' and 'y') describes a line in a two-dimensional space.
• The equation can be manipulated into slope-intercept form \(y = mx + b\), where 'm' is the slope and 'b' is the y-intercept.
• The solutions to a linear equation are all the points on its line; however, for systems, we seek the common solution point.

Linear equations are the building blocks of algebra. Understanding their characteristics, such as slope and y-intercept, helps in interpreting more complex equations and their graphs.

Simultaneous Equations

Simultaneous equations, or systems of equations, entail solving multiple equations with common variables.
We aim to find a solution that satisfies all given equations.When dealing with simultaneous equations:

• Each equation represents a condition that must be true at the same time, forming lines or curves in coordinate space.
• Solutions correspond to intersection points where these lines meet, representing values of variables that are consistent across all equations.
• Methods like substitution and elimination are commonly used to find these points, depending on the problem structure.

In our exercise, solving the system \(-\frac{2}{3} x+y =2\) and \(3 x-\frac{1}{2} y =4\), sought to find the 'x' and 'y' that simultaneously solve both equations, identified by their intersection point. Understanding simultaneous equations equips you to handle complex scenarios where several constraints apply together.