Problem 37 - Algebra A Combined Function

Question

Solve each system of linear equations by graphing. See Examples 3 through \(6 .\) \(\left\\{\begin{array}{l}6 x-y=4 \\ \frac{1}{2} y=-2+3 x\end{array}\right.\)

Step-by-Step Solution

Answer

The system of equations is dependent with infinitely many solutions, as both equations represent the same line.

Step 1: Rewrite Equations in Slope-Intercept Form

The first step is to rewrite each equation in the form of \(y = mx + b\). For the equation \(6x - y = 4\), solve for \(y\) to get \(y = 6x - 4\). For the second equation \(\frac{1}{2}y = 3x - 2\), multiply every term by 2 to eliminate the fraction and then solve for \(y\). This gives us the equation \(y = 6x - 4\).

Step 2: Recognize the Equations

Both equations simplify to the same line: \(y = 6x - 4\). This means they are not two separate lines but represent the same line graphically. Thus, the system of equations has an infinite number of solutions.

Step 3: Graph the Equations

Since both equations are the same, graph the line \(y = 6x - 4\) by plotting the y-intercept (-4) on the y-axis and then using the slope of 6 to locate the next point. For instance, from the point (0, -4), go up 6 units and right 1 unit to locate a second point at (1, 2).

Step 4: Conclusion from the Graph

Because both equations represent the same line, every point on the line is a solution. Therefore, the system has infinitely many solutions.

Key Concepts

Graphing Linear Equations

Understanding how to graph linear equations is a fundamental skill in algebra. When we take a linear equation and translate it onto the Cartesian plane, we essentially create a visual representation of all possible solutions for that equation. This visualization can help immensely in understanding the concept of solving systems of linear equations.

• To graph a linear equation, we need at least two points that lie on the line described by the equation.
• These points are typically found by identifying intercepts or selecting arbitrary x-values and solving for y.
• Once we've plotted the points, we draw a line that extends through both points; this is our graphed line.

Picturing these lines assists in identifying where multiple equations might intersect or overlap, providing visual insights into their solutions.

Slope-Intercept Form

The slope-intercept form of a linear equation is crucial because it provides a clear way to interpret and graph the equation. It is expressed as \(y = mx + b\), where:

• \(m\) is the slope of the line, indicating its steepness and direction.
• \(b\) is the y-intercept, representing where the line crosses the y-axis.
• This form allows for easy graphing and understanding of linear relationships.

In our exercise, both equations were eventually rewritten into the slope-intercept form of \(y = 6x - 4\). It shows that the lines have the same slope and intercept, meaning they coincide on a graph.

Infinite Solutions

Discovering infinite solutions in a system of equations occurs when the graphs of the equations coincide. That is, they are essentially the same line. In such a scenario:

• Every point on the line is a solution to the system, hence 'infinite solutions.'
• This occurs when, after simplifying the equations, they reduce to an identical line.
• No distinct intersection point exists because the entire line serves as the solution set.

In our problem, both equations resolved to \(y = 6x - 4\), resulting in infinite solutions because any point that satisfies one equation will naturally satisfy the other.

Solving Equations by Graphing

Solving equations by graphing is a method where we find the intersection point or points of two or more equations graphed on a coordinate plane. This technique illuminates potential solutions by visually identifying how lines interact:

• If lines intersect at a single point, that point provides the unique solution to the system.
• If lines are parallel and distinct, there are no solutions as they never meet.
• If lines overlap completely, there are infinite solutions, as is the case in our exercise.

Graphing provides a clear and intuitive approach to solving systems, reinforcing the algebraic work and offering insights into the relationships between linear equations.