Problem 9 - Thinking Mathematically

Question

Solve each system by graphing. Check the coordinates of the intersection point in both equations. \(\left\\{\begin{array}{l}y=x+5 \\ y=-x+3\end{array}\right.\)

Step-by-Step Solution

Answer

The solution to the system of equations is the point (-1, 4).

Step 1: Graph the equations

Begin by graphing both of the equations on the same coordinate system. The equation y = x + 5 is a line that intersects the y-axis at y = 5, and the line raises 1 unit for each unit moved to the right (its slope is 1). The line y = -x + 3 intersects the y-axis at y = 3, and the line falls 1 unit for each unit moved to the right (its slope is -1).

Step 2: Find the intersection point

Where the two lines cross is the solution to the system. For these two lines, the intersection occurs where x = -1 and y = 4.

Step 3: Check the solution

Verify the solution by substituting x = -1 and y = 4 into both original equations. Substituting into y = x + 5 gives: \(4 = -1 + 5\) which holds true. Substituting into y = -x + 3 gives: \(4 = 1 + 3\) which also holds true. Therefore, the solution has been confirmed.

Key Concepts

Graphing Method

The graphing method is a visual way to solve a system of equations. To use this method, you need to plot both equations on the same graph. This involves finding key points for each equation and drawing the lines that represent the solutions.

• First, identify the y-intercept of each equation. This is where the line crosses the y-axis.
• Next, use the slope of each line to determine the direction and steepness of the line.
• Plot at least two points for each line and connect them to extend the line across the graph.

Once both lines are plotted, the graph provides a visual display of potential solutions. The solution to the system is where both lines intersect. This method is especially useful for systems with distinct solutions that are easy to graph.

Intersection Point

The intersection point is a key concept when solving systems of equations graphically. It represents the coordinates \[(x, y)\] where the two lines meet on the graph.

• This point satisfies both equations simultaneously, meaning x and y coordinates plugged into both equations will be true.
• For our specific problem, the intersection point is at \( x = -1 \) and \( y = 4 \)

Finding the intersection can tell you if the system has:- One unique solution (the lines intersect at a single point).- No solution (the lines are parallel).- Infinitely many solutions (the lines are identical).

Linear Equations

Linear equations are fundamental elements in algebra, representing straight lines in geometry. A standard form for a linear equation is:\[y = mx + b\]where- \('m \)' is the slope of the line, determining its angle.- \('b \)' is the y-intercept, showing where the line crosses the y-axis.For the system in our exercise, the equations are linear: \[y = x + 5\] and \[y = -x + 3\]. These equations describe straight lines on the graph, and solving this system involves identifying where these two lines coincide. Linear equations are easy to work with due to their straightforward representation and predictable behavior. Understanding their graphical representation can simplify solving systems like the one given.

Coordinate Plane

The coordinate plane is a two-dimensional surface on which we can graph equations and visualize their solutions. It is characterized by a horizontal x-axis and a vertical y-axis, which intersect at the origin, \[(0,0)\].

• Each point on the plane can be described as a pair of coordinates \[(x, y)\]. These represent the horizontal and vertical positions, respectively.
• The coordinate plane allows us to graph linear equations by marking points that satisfy these equations and drawing lines through them.

Understanding how to navigate the coordinate plane is key to using the graphing method. It provides a backdrop for plotting the equations and ultimately finding the intersection point. The plane's grid-like structure makes it easier to accurately determine the exact location where lines intersect.