Problem 37 - Algebra and Trigonometry

Question

Use the graphical method to find all solutions of the system of equations, correct to two decimal places. \(\left\\{\begin{array}{l}{y=2 x+6} \\ {y=-x+5}\end{array}\right.\)

Step-by-Step Solution

Answer

The solution is \(x = -0.33\) and \(y = 5.33\).

Step 1: Write down the equations of the lines

The system of equations consists of two linear equations: \(y = 2x + 6\) and \(y = -x + 5\). These represent two lines on a coordinate plane.

Step 2: Identify key points for each line

For the first equation \(y = 2x + 6\), choose a couple of values for \(x\) to find corresponding \(y\) values. For example, if \(x = 0\), \(y = 6\); and if \(x = 1\), \(y = 8\). For the second equation \(y = -x + 5\), if \(x = 0\), \(y = 5\); and if \(x = 1\), \(y = 4\).

Step 3: Plot the points and draw the lines

Plot the points \((0, 6)\) and \((1, 8)\) for the first line, and draw the line through them. For the second line, plot the points \((0, 5)\) and \((1, 4)\), and draw the line through these points on the same coordinate plane.

Step 4: Identify the point of intersection

Observe the graph to identify where the two lines intersect. Approximate this point if necessary to two decimal places. The intersection is where the two equations return the same \(x\) and \(y\) values.

Step 5: Solve algebraically for accuracy

Since graphical estimation might lack precision, solve the equations algebraically to verify. Set \(2x + 6 = -x + 5\), rearrange to \(3x = -1\), giving \(x = -\frac{1}{3}\). Substituting \(x\) back into either equation, we find \(y = 2(-\frac{1}{3}) + 6 = \frac{16}{3}\).

Step 6: Present the solution

The graphical and algebraic methods indicate that the solution where the lines intersect is \(x = -0.33\) and \(y = 5.33\) when rounded to two decimal places.

Key Concepts

Graphical Method

The graphical method is a visual way to solve a system of equations by plotting each equation on a graph. The lines or curves you draw represent the relationships in the equations. By graphically representing these equations, you can visually find the solution by identifying where the graphs intersect.
This is particularly effective for finding approximate solutions quickly, as it leverages our natural ability to interpret visual information.

• Start by rewriting each equation in slope-intercept form, if necessary, so that it's easy to identify the slope and y-intercept.
• Identify a few key points that satisfy each equation.
• Plot these points on graph paper or a graphing tool, and draw the lines or curves through these points.
• The intersection point of the lines or curves represents the solution to the system of equations.

Graphical solutions are beneficial because they provide an intuitive understanding of how the equations relate to each other.

Linear Equations

Linear equations are fundamental building blocks in mathematics that describe a straight-line relationship between two variables, usually x and y. These are the simplest algebraic equations and are defined by their degree, which is one, meaning the highest power of the variable is 1.
Each linear equation can be expressed in the standard form: \[ax + by = c\] or the slope-intercept form: \[y = mx + c\], where m represents the slope and c is the y-intercept.

• The slope (m) indicates the steepness of the line and whether it is ascending or descending as it moves from left to right across the graph.
• The y-intercept (c) is the point where the line crosses the y-axis.

Understanding these components is crucial for graphing each equation in a system, which is the first step in employing the graphical method.

Coordinate Plane

The coordinate plane is a two-dimensional surface where you can plot points, lines, and curves. It’s defined by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical).
This plane is crucial for solving systems of equations using graphs. Each point on the plane can be identified by a pair of numbers written as (x, y), where x is the point's horizontal position and y is its vertical position on the graph.

• The region where these axes intersect is called the origin, indicated as (0, 0).
• The plane is divided into four quadrants, which help in identifying the sign and position of points.

Having a solid grasp of how the coordinate plane works is essential, as it forms the backdrop for graphing equations in the graphical method.

Intersection Point

The intersection point in the context of solving systems of equations is where the graphs of the equations meet on the coordinate plane. This point is crucial because it gives the exact solution to the system, reflecting the set of values that satisfy all equations in the system simultaneously.
For linear equations, the intersection point is typically a single point, as lines can either intersect at one spot, not at all (parallel lines), or be identical (overlap entirely, indicating infinitely many solutions).

• To find this point using the graphical method, plot each equation on the same coordinate plane and observe where the lines cross.
• The coordinates of this crossing point will be the solution to the system of linear equations.

This approach helps in understanding the solution geometrically, providing a clear and visually interpretable solution for where the variables of the equations align perfectly.