Problem 16 - Algebra 1: Concepts and Skills

Question

Graph the system of linear inequalities. $$ \begin{array}{r} {2 x-2 y \leq 6} \\ {x-y \leq 9} \end{array} $$

Step-by-Step Solution

Answer

The graphs of the inequalities \(2x - 2y \leq 6\) and \(x - y \leq 9\) are two lines intersecting in the plane, and the feasible region lies between these two lines, including the lines themselves.

Step 1: Rewrite the inequalities in slope-intercept form

The slope-intercept form of a line is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Rewrite the inequalities into this form to easily identify the slope and the y-intercept. The inequalities \(2x - 2y \leq 6\) and \(x - y \leq 9\) become \(y \geq x - 3\) and \(y \leq x - 9\) respectively.

Step 2: Graph the lines

Plot the lines \(y = x - 3\) and \(y = x - 9\) on the same graph. It's helpful to mark the y-intercepts \(-3\) and \(-9\) on the y-axis respectively and then use the slope (which is 1 for both lines) to find another point on the line. As \(y\) is greater or equal to \(x - 3\) and less than or equal to \(x - 9\), the line \(y = x - 3\) should be solid and upwards and the line \(y = x - 9\) should also be solid but downwards.

Step 3: Shade the feasible region

For the inequality \(y \geq x - 3\), shade the region above the line. For the inequality \(y \leq x - 9\), shade below the line. The feasible region, the solution to this system, is the area common to both inequations, i.e., between the two lines.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of writing the equation of a line so that the slope and y-intercept are easily recognizable. It is written as \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) represents the y-intercept, or the point where the line crosses the y-axis.

• The **slope** \(m\) is a measure of how steep the line is. Positive slope means the line is going upwards, while a negative slope means the line is going downwards.
• The **y-intercept** \(b\) is the value of \(y\) when \(x = 0\). It's the point where the line intersects the y-axis.

To use the slope-intercept form for a linear inequality, like in our original exercise, we rearrange the inequality to have \(y\) on one side. In our case, we transformed the inequalities \(2x - 2y \leq 6\) and \(x - y \leq 9\) into slope-intercept form to get \(y \geq x - 3\) and \(y \leq x - 9\). This format helps in identifying how the lines should be plotted on a graph, paving the way for an easy visualization of the solution set.

Feasible Region

The feasible region, when dealing with linear inequalities, represents the set of points that satisfy all the inequalities in a system. It is the overlapping area on the graph where the shaded regions of the inequalities meet.

• For the inequality \(y \geq x - 3\), the feasible region is any area above the line \(y = x - 3\).
• For the inequality \(y \leq x - 9\), the feasible region is any area below the line \(y = x - 9\).

The solution to the system of inequalities is thus the region that is covered by both inequalities. It's the shared area that fulfills both conditions. In our original example, this would be the space between the lines \(y = x - 3\) and \(y = x - 9\). This region includes all points \((x, y)\) that makes both inequalities true.
When graphing systems of linear inequalities, understanding the concept of a feasible region is crucial because it visualizes the possible solutions.

Plotting Graphs

Plotting graphs for linear inequalities involves several important steps. This visualization helps to see the various solutions that solve the inequalities.

• First, identify and draw the boundary lines. For \(y \geq x - 3\) and \(y \leq x - 9\), draw the lines \(y = x - 3\) and \(y = x - 9\).
• These lines often start with plotting the y-intercept, then use the slope to find another point, such as moving right 1 unit and up or down by the slope number.
• Mark the lines as solid or dashed depending on the inequality. Solid lines indicate \(\leq\) or \(\geq\), meaning points on the line are included in the solution. Dashed lines indicate \(<\) or \(>\), meaning points on the line are not included.
• Finally, shade the area that represents the solution to the inequality, always remembering to look where the shaded areas overlap for the feasible region.

By plotting the lines \(y = x - 3\) and \(y = x - 9\), and shading the appropriate regions, we visually represent the system of linear inequalities. This makes it easy to see and understand the set of solutions that satisfy the conditions.