Problem 24 - Algebra 1: Concepts and Skills

Question

Estimate the solution of the linear system graphically. Then check the solution algebraically. $$ \begin{array}{c} {5 x+6 y=54} \\ {-x+y=9} \end{array} $$

Step-by-Step Solution

Answer

After graphing and then determining our intersection point, we will insert this point back into our original equations to confirm the validity of our solution. Our final step is to cross-verify this solution by checking it against the observed intersection in the graph as well as via algebraic application.

Step 1: Graphing the Equations

Firstly, ensure that both equations are in the slope-intercept form \(y = mx + b\). Our equations becomes \(y = -\frac{5}{6}x + 9\) and \(y = x + 9\). Once they are in this format, graph the equations on the same graph paper or plotter.

Step 2: Finding the Intersection Point

The point of intersection of these equations on the graph represents the solution of the system. It is the pair of x and y values which satisfies both equations simultaneously.

Step 3: Checking the Solution Algebraically

Finally, take the coordinates of the intersection point obtained graphically and substitute them into the original equations. If they satisfy both equations, the solution is correct.

Step 4: Cross-validation

Cross-validate the estimated solution by observing the graphs and applying the solution algebraically. If both methods lead to the same result, the solution is most likely correct.

Key Concepts

Slope-Intercept Form

To solve a linear system of equations graphically, it's helpful to transform the equations into the slope-intercept form, which is: \[y = mx + b\] Here, \(m\) represents the slope of the line, showing how steep it is, while \(b\) is the y-intercept, where the line crosses the y-axis. By converting equations to this form, you make them easier to graph because you can clearly see their slope and starting point on the graph. For instance, the equation \(5x + 6y = 54\) is transformed by isolating \(y\) on one side to become \(y = -\frac{5}{6}x + 9\). This means:

• The slope \(m\) is \(-\frac{5}{6}\) - the line drops down by 5 for every 6 it goes across.
• The line meets the y-axis at 9.

This understanding aids in plotting the line accurately on a graph.

Graphing Equations

Once the equations are in the slope-intercept form, you can easily graph them. Begin by plotting the y-intercept (where the line touches the y-axis). For instance, if \(y = -\frac{5}{6}x + 9\), plot a point on 9 on the y-axis. Next, use the slope to find another point. Since the slope is \(-\frac{5}{6}\), from the initial point (9 on the y-axis), move 5 units down and 6 units to the right to locate the second point. Repeat the process for the second equation \(y = x + 9\). Start at point (9 on the y-axis) and use the slope \(1\) (or \(1/1\)) to find another point by moving 1 unit up for every 1 unit across. Connect the points for each equation to form two lines. This visual representation makes it straightforward to identify their intersection point, where they cross each other.

Intersection Point

The intersection point of the graphs of the equations represents the solution to the linear system. It's the spot where both lines meet on the graph, meaning the x and y values at this point satisfy both equations. For this exercise, once you have drawn both lines on the graph, look for the point where they intersect. It's a good idea to be precise when plotting and drawing straight, clear lines to avoid mistakes in identifying the intersection point. If it's tough to see exactly where they meet, it may help to draw more precise graphs on graph paper. This point is crucial because it gives you your solution: a pair \( (x, y) \), which needs to be verified algebraically.

Algebraic Solution

After estimating the solution graphically, it's essential to check it algebraically to confirm accuracy. Take the proposed coordinates of the intersection point from your graph and substitute these x and y values back into the original equations to see if both are satisfied. For example, if the graphical solution suggested the point \((x, y)\), substitute these values into both equations:

• For the equation \(5x + 6y = 54\), replace \(x\) and \(y\) with the coordinates and check whether both sides equal.
• Do the same for \(-x + y = 9\).

If the values satisfy both equations, the intersection point is indeed the correct solution. If the point does not satisfy one or both of the equations, re-evaluate your graphing or calculations. Confirming both methods—graphical and algebraic—align reflects the robustness of your solution.