Problem 36 - Precalculus with Limits

Question

Solving a System of Equations Graphically In Exercises \(33-44,\) solve the system graphically. $$\left\\{\begin{array}{rr}{-x+2 y=} & {-7} \\ {x-y=} & {2}\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is determined by the point of intersection of the two lines graphed.

Step 1: Set up the First Equation

Rewrite the first equation \( -x + 2y = -7 \) in the format of \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Isolate \( y \) as follows: \[2y = x - 7 -> y = 0.5x + 3.5\]where \( m = 0.5 \) and \( b = 3.5 \)

Step 2: Set up the Second Equation

Rewrite the second equation \( x - y = 2 \) in the format of \( y = mx + b \). Isolate \( y \) as follows: \[x - 2 = y -> y = x - 2\]where \( m = 1 \) and \( b = -2 \)

Step 3: Graph the two Lines

Graph the lines using their slopes and y-intercepts. Plot the first line with the slope 0.5 and y-intercept 3.5. Then, plot the second line with slope 1 and y-intercept -2. The point where the lines intersect is the solution.

Step 4: Identify the Intersection Point

The point at which the two lines intersect provides the solution to this system of equation. That's where both of these lines would satisfy both equations.

Key Concepts

Graphing Linear Equations

When we talk about graphing linear equations, we're referring to the process of drawing a line on a graph that represents all of the possible solutions to the equation. Each linear equation represents a straight line when graphed, and this line is defined by certain characteristics, which are its slope and y-intercept.

To graph a linear equation such as \( -x + 2y = -7 \), we need to rewrite it in slope-intercept form, which is \( y = mx + b \), where \( m \) stands for the slope of the line, and \( b \) represents the y-intercept, the point where the line crosses the y-axis.

Example:
For the equation \( -x + 2y = -7 \), after rearranging, we get \( y = 0.5x + 3.5 \). This means we can plot the line by starting at the y-intercept, which is 3.5, and then follow the slope, rising 0.5 units up for every 1 unit we move to the right.

Slope and Y-Intercept

The terms slope and y-intercept are fundamental to understanding linear equations. The slope, represented by \( m \), describes the steepness and direction of a line. It's calculated as the 'rise over run,' indicating how much the line goes up or down for a given distance across. A positive slope means the line is ascending, while a negative slope means it's descending.

The y-intercept, represented by \( b \), is the point where the line crosses the y-axis. It's the value of \( y \) when \( x \) is zero. Together, the slope and y-intercept define the position and angle of the line graphed on a coordinate plane.

For instance, in the equation \( y = x - 2 \), the slope is 1, implying that the line goes up by one unit for every unit it moves to the right. The y-intercept is -2, meaning the line crosses the y-axis at the point (0, -2).

System of Linear Equations

A system of linear equations consists of two or more equations with the same set of variables. The solution to such a system is the set of values that satisfies all equations simultaneously. In graphical terms, it's the point or points where the lines intersect.

For the given exercise, we have a system with two equations: \[\left\{\begin{array}{rr}{-x+2 y=} & {-7} \ {x-y=} & {2}\end{array}\right.\] These can be thought of as two different lines on the same graph. A solution to the system is any point that lies on both lines, which graphical methods help us find.

For more complex systems with more than two equations, the solution can be a line or a plane where all the equations' graphs intersect, but for two equations, it will generally be a single point unless the lines are parallel (no solution) or the same line (infinitely many solutions).

Intersection Point of Lines

The intersection point of lines is a crucial concept when solving a system of linear equations graphically. This point, where two or more lines cross, represents the values that satisfy all of the equations in the system at once. In a simple two-equation system, the intersection point is the pair of \( x \) and \( y \) coordinates that make both equations true.

In step 4 of our exercise, the intersection point provides the solution to the system. After plotting each line on a graph, we visually determine where they cross. This intersection represents the set of coordinates that solve both \( -x + 2y = -7 \) and \( x - y = 2 \).

Technology like graphing calculators or computer software can precisely pinpoint the intersection, but learning to do this by hand enhances understanding of the linear relationships and solution sets.