Problem 47 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to approximate all points of intersection of the graphs of equations in the system. Round your results to three decimal places. Verify your solutions by checking them in the original system. $$\left\\{\begin{array}{c} 7 x+8 y=24 \\ x-8 y=8 \end{array}\right.$$

Step-by-Step Solution

Answer

After graphing \`7x + 8y = 24\` and \`x - 8y = 8\` in a graphing utility, identify the intersection point of the two lines, round off the x and y coordinates to three decimal places, and check the solution by substituting these coordinates back into the original system of equations.

Step 1 - Graph the Equations

Enter the following equations into the graphing utility: \(7x + 8y = 24\) and \(x - 8y = 8\). These form two different lines.

Step 2 - Locate the Intersection

Observe the graph to locate the point where the lines intersect. This point represents the solution to the system of equations.

Step 3 - Round to Three Decimal Places

Depending on your graphing utility, you may be able to get the exact coordinates of the intersection. Round these values to three decimal places.

Step 4 - Check the Solution

Substitute the rounded-off coordinates for x and y back into the original equations to verify that they hold true.

Key Concepts

Graphing Utility

When tackling a system of equations, a graphing utility can be an invaluable tool. A graphing utility is a device or software that helps you plot equations on a coordinate plane. It visually represents equations as lines, curves, or other shapes, depending on the complexity. For systems of linear equations like the one provided,

• Enter each equation individually.
• Adjust the viewing window if necessary to see where they meet.
• Use functions available in the utility, such as "Trace" or "Find Intersection," to precisely determine where the graphs intersect.

By graphing the equations, you can quickly visualize potential solutions and better understand the relationship between the variables.

Point of Intersection

The point of intersection represents the solution to a system of linear equations graphically. It is the coordinate point

• where the graphs of the equations meet.
• that satisfies both equations simultaneously.

For the system given, when you graph the equations on a coordinate plane, you will look for a point where both lines cross.
This specific intersection is vital since it means that the values of \( x \) and \( y \) at this point work in both equations, satisfying them equationally. Using the graphing utility, the intersecting values are where the lines share a common solution.

Rounding Solutions

When graphing utilities provide solutions for equations, they may display long decimal values. In many practical situations, solutions need to be rounded for simplicity or clarity.

• Rounding to three decimal places means taking the solution to the thousandths.
• If the number following the last digit is 5 or more, increase the rounding digit by one.

Suppose the intersection point calculated is (1.63578, 2.79709). You will round them to (1.636, 2.797).
This rounding process is crucial in ensuring that the solutions are manageable and sufficiently precise for verification and further calculation.

Verification of Solution

After determining the rounded solution point, it's important to verify it by substituting back into the original equations. This checks whether the point really lies on both graphs.

• Take your rounded values of \( x \) and \( y \) and plug them into each equation one at a time.
• Solve the equations to confirm both sides are equal.

Verification is an essential step because it ensures accuracy. For example, if you substitute \( x = 1.636 \) and \( y = 2.797 \) into both equations, both left-hand sides should equal the right-hand sides if the point is correct.
This step validates that the graphical-method-derived solution satisfies the algebraic requirements of the system.