Problem 64 - Intermediate Algebra

Question

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.) $$ \left\\{\begin{array}{l} 4 x=3 y-1 \\ 3 y=4-8 x \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \( \left( \frac{1}{4}, \frac{2}{3} \right) \); the system is consistent and independent.

Step 1: Rewriting the System in Slope-Intercept Form

First, let's rewrite each equation in the system in the slope-intercept form, which is \( y = mx + b \).\For the first equation:\[ 4x = 3y - 1 \]Add 1 to both sides:\[ 4x + 1 = 3y \]Divide everything by 3 to solve for \( y \):\[ y = \frac{4}{3}x + \frac{1}{3} \]For the second equation:\[ 3y = 4 - 8x \]Divide everything by 3:\[ y = -\frac{8}{3}x + \frac{4}{3} \]

Step 2: Plotting the Lines on the Graph

Using the slope-intercept form from Step 1, plot the following equations:For the first line, \( y = \frac{4}{3}x + \frac{1}{3} \): - The y-intercept is \( \frac{1}{3} \) (point \( (0, \frac{1}{3}) \)).- The slope is \( \frac{4}{3} \), meaning for every right movement of 3 units, move up 4 units.For the second line, \( y = -\frac{8}{3}x + \frac{4}{3} \):- The y-intercept is \( \frac{4}{3} \) (point \( (0, \frac{4}{3}) \)).- The slope is \( -\frac{8}{3} \), meaning for every right movement of 3 units, move down 8 units.

Step 3: Identifying the Point of Intersection

Examine where the two lines intersect on the graph:From plotting, you will see that the point of intersection can be deduced by finding where both equations yield the same \(x\) and \(y\). To do this algebraically, set the two expressions for \(y\) from Step 1 equal:\[ \frac{4}{3}x + \frac{1}{3} = -\frac{8}{3}x + \frac{4}{3} \]Solve for \(x\):Add \( \frac{8}{3}x \) to both sides:\[ \frac{4}{3}x + \frac{8}{3}x = \frac{4}{3} - \frac{1}{3} \]Combine like terms:\[ 4x = 1 \]Solve for \(x\):\[ x = \frac{1}{4} \]Substitute \( x = \frac{1}{4} \) back into either equation to solve for \( y \):Using \( y = \frac{4}{3}x + \frac{1}{3} \):\[ y = \frac{4}{3} \times \frac{1}{4} + \frac{1}{3} = \frac{1}{3} + \frac{1}{3} = \frac{2}{3} \]Thus, the intersection point is \( \left( \frac{1}{4}, \frac{2}{3} \right) \).

Step 4: Conclusion about the System

Since the two lines intersect at one point, the system is consistent and independent. The solution to the system is the point of intersection.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of expressing a linear equation as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) represents the y-intercept. The slope tells us how steep the line is, indicating how many units you go up or down (rise) for each unit you move to the right (run). The y-intercept is where the line crosses the y-axis.
Processing a linear equation into this form allows us to easily graph the function, as it clearly shows both the slope and the starting point on the y-axis. To transform an equation into this form, we need to solve for \( y \) and isolate it on one side of the equation. This transformation is crucial in graphing systems of equations because it allows us to visually interpret where two lines intersect, demonstrating potential solutions for the system.

Consistent and Independent Systems

When graphing systems of equations, it's important to classify them based on their solutions. A consistent system is one that has at least one solution. An independent system is one where the two equations represent two different lines that intersect at exactly one point.
In the provided solution, our system was categorized as consistent and independent. This means the two lines we graphed have different slopes, intersecting at a single point, which represents the solution. As opposed to dependent systems where lines overlap (infinite solutions) or inconsistent systems where lines are parallel (no solutions), consistent and independent systems give us the unique point where both equations are satisfied simultaneously.

Point of Intersection

The point of intersection of two lines is crucial because it represents the solution to a consistent and independent system of equations. It is the coordinate point \((x, y)\) where the graphs of these equations cross. For the system given in the exercise, this point was calculated to be \( \left( \frac{1}{4}, \frac{2}{3} \right) \).
To find this point, you either graph the equations to see where they meet visually or solve algebraically by setting the expressions for \( y \) equal to one another. This involves finding the value for \( x \) where both equations coincide. Once we have \( x \), substituting it back into one of the original equations gives us the corresponding \( y \) value, allowing us to fully determine the intersection's coordinate. Understanding this process is key to solving systems of linear equations comprehensively.