Problem 90 - Precalculus

Question

Apply a graphing utility to graph the two equations \(-3 x+7 y=2\) and \(6 x-14 y=3 .\) Approximate the solution to this system of linear equations.

Step-by-Step Solution

Answer

The system of equations has no solution because the lines are parallel.

Step 1: Convert Equations to Slope-Intercept Form

Start by converting both equations to the slope-intercept form, which is \[ y = mx + b \]. For the first equation, \(-3x + 7y = 2\), solve for \(y\): Add \(3x\) to both sides to get:\[ 7y = 3x + 2 \]Now, divide every term by \(7\):\[ y = \frac{3}{7}x + \frac{2}{7} \].For the second equation, \(6x - 14y = 3\), solve for \(y\):Add \(-6x\) to both sides:\[-14y = -6x + 3\]Divide every term by \(-14\):\[ y = \frac{3}{7}x - \frac{3}{14} \].

Step 2: Graph the Two Lines

Use graphing software or a graphing calculator to plot the lines represented by the equations. The first line's equation is \[ y = \frac{3}{7}x + \frac{2}{7} \] and the second line is \[ y = \frac{3}{7}x - \frac{3}{14} \]. The lines will have identical slopes, indicated by \( \frac{3}{7} \), but different y-intercepts, \( \frac{2}{7} \) and \(-\frac{3}{14} \), respectively. This means they are parallel.

Step 3: Identify the Solution

Since parallel lines never intersect, this system of equations has no points of intersection. Therefore, there is no solution to the system of equations.

Key Concepts

Graphing Lines

Graphing lines is an essential skill when working with linear equations. By plotting the equation of a line on a graph, we visually interpret the relationship between variables. The line represents all possible solutions of the equation. Graphing gives us a tangible understanding of the behavior of the equation.
To graph a line, you typically need the equation in one of its forms, such as the slope-intercept form. Then, you identify some key points to plot. These could be the y-intercept and a point derived from the slope. These two points guide the drawing of the entire line. Using graphing utilities, like calculators or software, enhances accuracy.

• First, rearrange the equation to identify the slope (rate of change) and the y-intercept (starting point).
• Plot the y-intercept on the y-axis.
• From this point, use the slope to find a second point by rising or falling and running horizontally, based on the fraction defining the slope.
• Draw a line through these points, extending it to cover the graph area.

Graphing helps solve systems of equations by showing where lines meet, representing the solution of the system, if any.

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most user-friendly forms for graphing. Written as \( y = mx + b \), it quickly reveals two critical pieces of information: the slope \( (m) \) and the y-intercept \( (b) \).
The slope \( (m) \) indicates how steep the line is, describing the rate of change between the variable on the x-axis and the variable on the y-axis. For example, a slope of \( \frac{3}{7} \) moves up three units for every seven units it moves to the right.
The y-intercept \( (b) \) shows where the line crosses the y-axis. This point occurs when \( x = 0 \). Recognizing a y-intercept in the form lets you instantaneously pinpoint the starting point for drawing the line.
By converting equations into slope-intercept form, as described in the step-by-step solution, you simplify the graphing process. You go from algebraic manipulation straight to visual representation, making it easier to comprehend and communicate the relationships described by the equations.

Parallel Lines

Parallel lines are a fascinating concept in geometry and algebra. These lines run side-by-side and never intersect, no matter how far they extend. For lines on a coordinate plane to be parallel, they must have the same slope. This means they rise and run at identical rates, maintaining a constant distance apart.
In the context of systems of linear equations, if the lines are parallel, the system has no solutions because there's no point of intersection. Looking at the step-by-step solution of the exercise, the equations \( y = \frac{3}{7}x + \frac{2}{7} \) and \( y = \frac{3}{7}x - \frac{3}{14} \) showcase this property. Both lines have slopes of \( \frac{3}{7} \), confirming their parallel nature.
Understanding parallel lines aids in visualizing whether a system of equations is solvable. If lines are parallel, they do not meet, hence indicate the system of equations is inconsistent. When working on linear equations, knowing about parallel lines can save time by revealing system properties graphically without needing further calculations.