Algebra 2 by Richard Wright

Previous LessonTable of Contents Next Lesson

Are you not my student and
has this helped you?

1-01 Solve Linear Systems of Equations and Inequalities by Graphing

Mr. Wright teaches the lesson.

Objectives:

SDA NAD Content Standards (2018): AII.5.3, AII.6.1, AII.6.5, AII.7.1

tacos
Figure 1: Tacos. (Pixabay/adamlot)

Mike and Sally went to a fast-food restaurant and each ordered burritos and tacos. Mike ate one of Sally's tacos and wants to pay her for it. However, they do not have receipts and have left the restaurant so that they do not have the menu prices. All they know are the total paid by each person and the amount of food that was ordered. This is actually enough information to find out how much a taco cost by solving a system of equations.

Systems of Equations

A system of equations is several equations with the same solution. Since the equations all have the same solution, they must all go through that point on a graph. Solutions are ordered pairs that give a true statement in all the equations in the system.

Example 1

Determine if (3, −4) is a solution to \(\left\{ \begin{align} 4x+y &=8 \\ 2x-3y &=18 \end{align} \right.\).

Solution

Substitute the point into both equations.

4x + y = 8

4(3) + (−4) = 0

12 − 4 = 8

8 = 8 ✔

Now try the other equation.

2x − 3y = 18

2(3) − 3(−4) = 18

6 + 12 = 18

18 = 18 ✔

The point gives a true statement when substituted into both equations, so it is a solution to the system.

Try It 1

Determine if (−2, 1) is a solution to \(\left\{ \begin{align} 3x+2y & =-4 \\ x+3y & =1 \end{align} \right.\).

Answer

Yes

Solve Systems by Graphing

Because the solution to a system of equations is an ordered pair that is a solution to each equation in the system, the graphs of each equation must go through the same point.

Solve a System by Graphing

To solve a system by graphing,

  1. Graph all the equations on the same coordinate plane. See lesson 0-05.
  2. The solutions are all the points of intersection.

Example 2

Solve the system by graphing: \(\left\{ \begin{align} 8x-y & =8 \\ 3x+2y & =-16 \end{align} \right.\).

Solution

Graph each equation by writing them in slope-intercept form (solve for y).

$$ \left\{ \begin{align} y &= 8x-8 \\ y &= -\frac{3}{2} x-8 \end{align} \right. $$

Figure 2: The intersection of 8x − y = 8 and 3x + 2y = −16 is (0, −8).

The solution is the point of intersection, (0, −8).

Try It 2

Solve the system by graphing: \(\left\{ \begin{align} -2x+y & =5 \\ y & =-x+2 \end{align} \right.\).

Answer

The solution is (–1, 3).

Example 3

Solve the system by graphing: \(\left\{ \begin{align} y & = \frac{2}{3}x+1 \\ 2x-3y & =-3 \end{align} \right.\).

Answer

Graph the equations by writing them in slope-intercept form (solve for y).

$$ \left\{ \begin{align} y=\frac{2}{3} x+1 \\ y=\frac{2}{3}x+1 \end{align} \right. $$

Figure 3: The lines are the same.

The are infinitely many solutions.

Try It 3

Solve the system by graphing: \(\left\{ \begin{align} y=-\frac{1}{2}x+2 \\ y=-\frac{1}{2}x-3 \end{align} \right.\).

Answer

The lines do not intersect, so there is no solution.

Finding Points of Intersection on a TI-84
TI-84
Figure 4: TI-84 Point of Intersection
  1. Push Y= and input the functions, one per line.
  2. Push GRAPH and zoom until the points of intersection are shown.
  3. Push 2ND CALC and select intersect.
  4. The calculator asks for the First curve, so push ENTER to select one of the curves.
  5. The calculator asks for the Second curve. The calculator automatically moved the cursor to the next curve, so push ENTER to select the other curve. If needed, the up or down arrows will switch the cursor between the curves.
  6. The calculator asks for a guess. Use the left and right arrows to move the cursor near the point of intersection and press ENTER.
  7. Read the point off the graph. The solution is (0.364, −0.091).
Finding Linear Regression on a NumWorks graphing calculator
NumWorks
Figure 5: NumWorks Point of Intersection
  1. On the home screen select Grapher.
  2. In the Expressions tab, enter all the equations.
  3. Move to the Graph tab.
  4. Press OK.
  5. Select Intersection.
  6. Read the points off of the screen. The solution is (0.364, −0.091).

Example 4

Use a graphing calculator to find the solution of \(\left\{ \begin{align} y & = \frac{2}{3} x - \frac{1}{3} \\ y & = -3x + 1 \end{align} \right.\).

Solution on TI-84

On a TI-84 calculator,

  1. Press the Y= button and input the two equations, one in Y1 and one in Y2.
    2. Figure 6: Y= screen on TI-84
  2. Press the GRAPH and make sure that the point of intersection is visible. If not, use the ZOOM features to get the make the point of intersection visible.
    3. Figure 7: Graph on TI-84
  3. Push 2ND CALC and select intersect.
  4. The calculator asks for the First curve, so push ENTER to select one of the curves.
  5. The calculator asks for the Second curve. The calculator automatically moved the cursor to the next curve, so push ENTER to select the other curve. If needed, the up or down arrows will switch the cursor between the curves.
  6. The calculator asks for a guess. Use the left and right arrows to move the cursor near the point of intersection and press ENTER.
  7. Read the point off the graph.
    8. Figure 8: Solution on TI-84

Solution on NumWorks

On the NumWorks calculator,

  1. On the home screen select Grapher.
  2. In the Expressions tab, enter all the equations.
    3. Figure 9: Expressions tab on NumWorks
  3. Move to the Graph tab.
    4. Figure 10: Graph tab on NumWorks
  4. Press OK.
  5. Select Intersection.
  6. Read the points off of the screen.
    7. Figure 11: Solution on NumWorks

Systems of Inequalities

Inequalities usually have many solutions. It is usually easiest to describe the solution to an inequality in two variables by the shaded area on a graph. The same is true for a system of inequalities.

Solve a System of Inequalities

To solve a system of inequalities in two variables,

  1. Graph all the inequalities on the same graph. See lesson 0-07.
  2. The solution is the area where the shaded areas of each inequality overlap.

Example 5

Solve the system of inequalities.

$$ \left\{ \begin{align} 2x+y & < -2 \\ x-y & ≥ -3 \end{align} \right. $$

Solution

Write the inequalities in slope-intercept form.

$$ \left\{ \begin{align} y & < -2x-2 \\ y & ≤ x+3 \end{align} \right. $$

Graph the inequalities.

Figure 12: The solutions are the intersection of the shaded areas.

The solution are all the points where the shaded areas overlap.

Try It 4

Solve the system of inequalities: \(\left\{ \begin{align} y & ≥ 12x-2 \\ x+y & ≤ 4 \end{align} \right.\)

Answer

The solution are all the points where the shaded areas overlap.

Practice Exercises

    Graph the system and estimate the solution.

  1. \(\left\{ \begin{align} y & = -3x+2 \\ y & =2x-3 \end{align} \right.\)
  2. \(\left\{ \begin{align} y=-x+3 \\ -x-3y=-1 \end{align} \right.\)
  3. \(\left\{ \begin{align} y=2x-10 \\ x-4y=5 \end{align} \right.\)
  4. \(\left\{ \begin{align} y & =3x+2 \\ y & =3x-2 \end{align} \right.\)
  5. \(\left\{ \begin{align} y=2x-1 \\ -6x+3y=-3 \end{align} \right.\)

    6. Graph the system of inequalities.

  6. \(\left\{ \begin{align} 2x-y & > 4 \\ x-2y & < -1 \end{align}\right.\)
  7. \(\left\{ \begin{align} 5x-4y & ≤ 3 \\ 3x+2y & ≤ 15 \end{align} \right.\)
  8. \(\left\{ \begin{align} y & ≤ 2 \ x+3 \\ y & ≥ 5 \end{align}\right.\)
  9. \(\left\{ \begin{align} y < \frac{2}{3}x+2 \\ 2x-3y < 6 \end{align} \right.\)

    10. Problem Solving

  10. Sally and Jamie are buying some little books about Jesus to pass out around their neighborhood. The prices are not labeled, but Sally bought 3 copies of book A and 4 copies of book B and paid $18. Jamie bought 2 copies of book A and 3 copies of book B and paid $13. How much does each book cost? (Solve by graphing.)

    11. Mixed Review

  11. (0-08) Tell whether the data has positive, negative, or approximately no correlation.
    x10.521.50.7532.5
    y1.881.941.751.811.911.631.69
  12. (0-06) Describe the transformations: \(g(x)=\frac{1}{3}\left|x\right|+5\)
  13. (0-04) Find the equation of the line with slope \(-\frac{2}{3}\) and passes through (2, −1).
  14. (0-03) Solve \(3\left|x-4\right|=12\).
  15. (0-01) Solve for y: 16xy + 2y = 1

Answers

  1. (1, −1)
  2. (4, −1)
  3. (5, 0)
  4. No solution
  5. Infinitely many solutions
  10. Book A: $2, Book B: $3
  11. Negative
  12. Vertical shrink by factor of \(\frac{1}{3}\), shift up 5
  13. \(y=-\frac{2}{3}x+\frac{1}{3}\)
  14. 0, 8
  15. \(y=\frac{1}{16x+2}\)