Problem 20 - Algebra 1: Concepts and Skills

Question

Estimate the solution of the linear system graphically. Then check the solution algebraically. $$ \begin{aligned} 5 x+4 y &=16 \\ y &=-16 \end{aligned} $$

Step-by-Step Solution

Answer

The system of equations has no solutions.

Step 1: Graph the Equations

Start by drawing each equation on graph paper. The equation \( 5x + 4y = 16 \) is a straight line that can be written in the form \( y = -1.25x + 4 \). Plot the y-intercept at 4 and with a slope of -1.25, plot the following points and draw the line. The equation \( y = -16 \) is a horizontal line crossing the y-axis at -16. Plot the line along -16 on the y-axis.

Step 2: Determine the solution graphically

The solution to the system of equations is the point where both lines intersect. In this case, it can be seen that the lines don't intersect. This suggests that the system of equations has no solutions.

Step 3: Check the Solution Algebraically

To confirm, plug the graphical solution, which is non-existent in this case, into both equations. Since there is no point of intersection, there's no feasible solution for the given equations.

Key Concepts

Graphical Method

Visualizing equations helps us understand their relationships. The graphical method involves drawing the lines of each equation on a graph to find their intersection point. This intersection represents the solution of the system of equations, assuming they intersect.

- **Equation 1**: Start with the equation \(5x + 4y = 16\). It can be rearranged into \(y = -1.25x + 4\). Here, “4” is the y-intercept, meaning the line crosses the y-axis at this point. The slope, \(-1.25\), tells us how steep the line is. For every step right, the line goes 1.25 steps down.

- **Equation 2**: The equation \(y = -16\) represents a simple horizontal line. This line sits along the y-axis at -16, meaning it doesn’t rise or fall, so its slope is 0.

When you graph these lines, look for an intersection. If the lines never meet, there’s no solution. As in our example, these lines don't intersect, showing that this system has no solutions.

Algebraic Verification

Algebraic verification confirms what we find graphically. It involves solving the system using algebra to check whether an intersection exists. This step is crucial to validate the graphical method, especially if the graph is drawn manually or has estimation inaccuracies.

- **Substitute and Solve**: With the equations \(5x + 4y = 16\) and \(y = -16\), substitute \(y = -16\) into the other equation, transforming it into \(5x + 4(-16) = 16\) leading to \(5x - 64 = 16\).

- **Simplify**: Solve for \(x\) by adding 64 to both sides, giving \(5x = 80\). Then divide by 5 to find \(x = 16\). If this \(x\) value led to a contradiction, it would suggest there is no solution. In our case, the graphical no-intersection scenario aligns with our algebraic verification, confirming no solutions.

Linear Equations

Linear equations like the ones we worked with are foundations of algebra. They involve variables with no exponents (only to the first power), usually creating straight lines when graphed.

- **Key Components**: A linear equation in the form \(ax + by = c\) involves constants \(a\), \(b\), and \(c\). These equations can be represented in the slope-intercept form \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept.

- **Systems of Linear Equations**: Often, problems involve finding where two equations intersect. These intersections can show us one solution, many solutions, or none. Parallel lines, which never intersect, indicate no solutions, as seen in this example.

Linear equations thus help solve practical problems like finding costs, speeds, or ages through their straightforward mathematical relationships, making them a powerful tool in math.