Problem 9 - Elementary Algebra Within Reach

Question

In Exercises \(5-10\), solve the system by graphing. $$ \left\\{\begin{array}{l} x-y=3 \\ x+y=3 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system of equations is the point (1, 2).

Step 1: Transform the Equations into Slope-Intercept Form

Express each equation in the form \(y = mx+b\), where m is the slope and b is the y-intercept. So the given equations \(x - y = 3\) and \(x + y = 3\) will be rearranged as \(y = x - 3\) and \(y = -x + 3\) respectively.

Step 2: Graph the Equations

Plot both equations on the same graph. From the slope-intercept forms, one can see that the equation \(y = x - 3\) has a slope of 1 and a y-intercept (when x=0) of -3. Consequently, it crosses the y-axis at the point (0,-3). Similarly, the equation \(y = -x + 3\) has a slope of -1 and a y-intercept (when x=0) of 3, so it crosses the y-axis at the point (0,3). Draw these lines.

Step 3: Identify the Intersection Point

The point where both lines cross is the solution to the system of equations. In this case, they intersect at the point (1, 2).

Key Concepts

Graphing Method

The graphing method is a visual way of solving a system of equations. It involves drawing the equations on a coordinate plane to see where they intersect.
This method is especially handy when dealing with linear equations, as it allows us to easily identify solutions at the point where two lines meet.

• Start by transforming your equations into a more manageable form, typically the slope-intercept form.
• Plot each line on a graph using their slopes and y-intercepts.
• Look for the intersection point which represents the solution.

Graphing helps in understanding the relationship between the equations and visually demonstrates their solutions. It's important to be accurate with plotting to ensure the intersection point is correct.

Slope-Intercept Form

The slope-intercept form is a specific way to write the equation of a line. It's given by the formula: \[ y = mx + b \] where:

• \(m\) represents the slope of the line, indicating how steep the line is.
• \(b\) is the y-intercept, showing where the line crosses the y-axis.

This form is particularly useful for graphing because it provides immediate information about the line's slope and where to start plotting (the y-intercept).Converting equations into slope-intercept form makes it easier to graph and compare different lines, essential for solving systems of equations through the graphing method.

Intersection Point

The intersection point is crucial when solving systems of equations graphically. It is where two or more lines cross each other on a graph.
In a system of linear equations, the intersection point represents the solution because it satisfies both equations simultaneously.
For example, if we have the equations:

• \(y = x - 3\)
• \(y = -x + 3\)

The lines intersect at the point (1, 2). This means substituting (1, 2) back into both equations should hold them true, confirming it as the solution.
Finding the intersection point graphically is intuitive as it shows the exact values for the variables that solve both equations.

Linear Equations

Linear equations are mathematical expressions that graph as straight lines on the coordinate plane. A typical linear equation in two variables is written as:\[ ax + by = c \]where \(a\), \(b\), and \(c\) are constants.
These equations are called 'linear' because their graphs are straight lines. The concept is central to algebra and particularly important when solving systems of equations.
Systems of linear equations involve finding values that satisfy multiple such expressions at once. They are often solved using methods like substitution, elimination, or graphing.
Understanding linear equations and their properties is fundamental when dealing with graphs and intersections within mathematical models and real-life applications.