Problem 49 - College Algebra with Corequisite Support

Question

For the following exercises, use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth. $$ \begin{aligned} 0.15 x+0.27 y &=0.39 \\ -0.34 x+0.56 y &=1.8 \end{aligned} $$

Step-by-Step Solution

Answer

The solution is the intersection point of the lines, rounded to the nearest hundredth.

Step 1: Identify Equations

We have two equations: \( 0.15x + 0.27y = 0.39 \) and \( -0.34x + 0.56y = 1.8 \). These equations represent two lines on a graph.

Step 2: Graph the Equations

Use a graphing device to plot the equations \( y = -\frac{0.15}{0.27}x + \frac{0.39}{0.27} \) and \( y = \frac{0.34}{0.56}x + \frac{1.8}{0.56} \). Simplify these to slope-intercept form for accurate graphing.

Step 3: Find the Intersection

Use the intersect function on your graphing device to find where the two lines intersect. This point of intersection represents the solution \((x, y)\) of the system of equations.

Step 4: Round the Intersection Coordinates

If the graphing device provides an intersection point, round the \(x\) and \(y\) coordinates to the nearest hundredth for each value.

Key Concepts

Graphing Linear Equations

To understand systems of linear equations like the ones given, it's helpful to graph them. Graphing linear equations involves plotting the line represented by each equation on the same coordinate plane. To do this effectively:

• Convert the standard form of the equations to slope-intercept form, such as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. This makes it easier to graph.
• For example, take the equation \(0.15x + 0.27y = 0.39\) and rewrite it as \(y = -\frac{0.15}{0.27}x + \frac{0.39}{0.27}\).
• Simplify to obtain the slope (\(m\)) and y-intercept \((b)\).

These transformations help clearly denote how steep the line is and where it crosses the y-axis, both essential for accurate graphing on a graphing calculator.

Intersection Point

The intersection point of two lines on a graph represents the exact solution to a system of linear equations.

• This point is where the two equations share the same \(x\) and \(y\) values.
• By finding this point, you discover the coordinates that satisfy both equations simultaneously.

For our particular system, after graphing both lines, you use the intersect function on a graphing calculator to find where they cross. This intersection point is crucial as it gives you the answer to the system of equations you are working with.

Graphing Calculator

A graphing calculator is an indispensable tool for solving systems of equations, especially when dealing with more complex or non-integer solutions. Here's how to make the most of it:

• Input each equation into the calculator, ensuring you’ve solved for \(y\), making them easier to graph.
• Use the calculator’s graphing function to plot each line accurately on the coordinate plane.
• Utilize the intersect function to find the point where the two lines meet.
• This function calculates the precise \(x\) and \(y\) coordinates where both equations are true.

The graphing calculator not only visualizes the solution but allows for an efficient and effective approach to solving equations.

Rounding Solutions

In mathematical computations, rounding solutions to a certain decimal place provides a cleaner and more manageable form of an answer. For the current exercise, rounding to the nearest hundredth is specified:

• Once you have the intersection point from the graphing calculator, check the coordinates.
• Round the \(x\) and \(y\) values to two decimal places. For instance, if your calculator shows \((1.234, 4.567)\), you round it to \((1.23, 4.57)\).

Rounding to the nearest hundredth is particularly useful when dealing with real-world data that requires a balance between precision and simplicity. It helps in ensuring that the solutions are practical while maintaining a degree of accuracy necessary for analysis.