Additional Need To know Notes

17: Solving an Equation for a Variable

Example: Solve for x in the equation 2x + 3 = 11.Steps:

1. Subtract 3 from both sides:2x = 8

2. Divide both sides by 2:x = 4

18: Graphs of Equations with Two Variables

Example: Graph the equation y = 2x + 1.Steps:

1. Create a table of values (e.g., x: -1, 0, 1, y: -1, 1, 3).

2. Plot the points (-1, -1), (0, 1), and (1, 3) on a coordinate plane.

3. Draw a straight line through the points.

19: Graphs of Equations with Three Variables

Example: Graph the equation z = x + y.Steps:

1. Choose values for x and y (e.g., x: 0, 1, y: 0, 1).

2. Calculate z for each pair (e.g., (0,0) -> z=0, (1,0) -> z=1, (0,1) -> z=1, (1,1) -> z=2).

3. Plot the points in a 3D coordinate system (x, y, z).

4. Connect the points to form a plane.

20: Solving a System of Equations

Example: Solve the system:

1. x + y = 5

2. x - y = 1Steps:

3. From the first equation, express y in terms of x: y = 5 - x.

4. Substitute this expression into the second equation:x - (5 - x) = 1.

5. Simplify:2x - 5 = 1

6. Add 5 to both sides:2x = 6

7. Divide by 2:x = 3

8. Substitute x back into y = 5 - x:y = 2Solution: (3, 2)Schedule Final Review.

How to Graph Linear Equations from Word Problems

Steps to Solve and Graph:

1. Read the Problem Carefully: Understand what is being asked and identify the variables involved.

   • Example: A company sells t-shirts for $20 each. Let x be the number of t-shirts sold, and y be the total revenue.

2. Write the Equation: Translate the word problem into a mathematical equation based on relationships.

   • For the example: Revenue = Price × Quantity leads to the equation: y = 20x.

3. Create a Table of Values: Choose several values for x (independent variable) and calculate corresponding y values (dependent variable).

   • Example Table (for y = 20x):

     x	y
     0	0
     1	20
     2	40
     3	60

4. Plot the Points: Locate the points from your table on a coordinate plane.

   • Plot (0, 0), (1, 20), (2, 40), and (3, 60).

5. Draw the Line: Connect the points with a straight line, as linear equations represent a straight line on a graph.

6. Interpret the Graph: Analyze what the linear graph represents in terms of the problem.

   • In this case, the slope indicates that for every t-shirt sold, total revenue increases by $20.

Additional Example:

Problem: A cell phone company charges a monthly fee of $30 plus $0.10 per minute for calls.

• Variables: Let x = minutes used, y = total charge.

• Equation: y = 0.10x + 30.

Steps:

1. Create a table of values:

   • For x = 0, y = 30.

   • For x = 100, y = 30 + (0.10 × 100) = 40.

   • For x = 200, y = 30 + (0.10 × 200) = 50.

2. Plot the points: (0, 30), (100, 40), (200, 50).

3. Draw the line connecting the plotted points and interpret the graph accordingly.

Graphing Linear Equations with Three Variables using Word Problems

1. Understanding the Problem: Read the problem carefully to identify the three variables involved.

   • Example Problem: A bakery produces bread (x), cakes (y), and cookies (z). The total number of baked goods produced is 100 per day. If they produce twice as many cakes as cookies, how many of each item do they make?

2. Define the Variables:

   • Let x = number of breads

   • Let y = number of cakes

   • Let z = number of cookies

3. Write the Equations Based on the Problem:

   • Total goods equation: x + y + z = 100

   • Relationship between cakes and cookies: y = 2z

4. Choose Values for Two Variables:

   • Choose simple values for z (cookies) first, then calculate x and y.

   • Example Choices: 0, 10, 20 (for cookies)

5. Calculate the Third Variable:

   • For z = 0:

     • y = 2(0) = 0,

     • x + 0 + 0 = 100 → x = 100 → (100, 0, 0)

   • For z = 10:

     • y = 2(10) = 20,

     • x + 20 + 10 = 100 → x = 70 → (70, 20, 10)

   • For z = 20:

     • y = 2(20) = 40,

     • x + 40 + 20 = 100 → x = 40 → (40, 40, 20)

6. Plot the Points in 3D Coordinate System:

   • Plot the points (100, 0, 0), (70, 20, 10), (40, 40, 20) in a 3D graph to visualize the relationships.

7. Connect the Points:

   • Connect the points to form a plane representing the relationship of the baked goods produced in the bakery.

8. Interpret the Graph:

   • Analyze what the 3D representation means for the bakery's production limits and options based on chosen quantities for cookies.

To solve the system of equations:

1. y = 9 + 3z

2. y = 51 - 3z

Step 1: Set the equations equal to each other since both equal y.

• 9 + 3z = 51 - 3z

Step 2: Solve for z.

• Add 3z to both sides:

• 9 + 6z = 51

• Subtract 9 from both sides:

• 6z = 42

• Divide by 6:

• z = 7

Step 3: Substitute z back into one of the original equations to find y.

• Using the first equation:

• y = 9 + 3(7)

• y = 9 + 21

• y = 30

Final Solution: z = 7, y = 30.

The value of x is not provided in these equations. If there were an equation related to x, we could find its value as well. But with the given equations, we can only determine z and y.

To solve the system of equations:

1. 9a + 3b = 30

2. 8a + 4b = 28

Step 1: Simplify the equations if possible.

• Divide the first equation by 3:a + b = 10

• Divide the second equation by 4:2a + b = 7

Step 2: Solve for b in terms of a using the first equation:b = 10 - a

Step 3: Substitute b into the second equation:2a + (10 - a) = 72a + 10 - a = 7a + 10 = 7

Step 4: Solve for a:a = 7 - 10a = -3

Step 5: Substitute a back into the equation for b:b = 10 - (-3)b = 10 + 3b = 13

Final Solution:a = -3b = 13.

To solve for q in the equation p = 6 - 2q:

1. Rearrange the equation to isolate q:

   p + 2q = 6

2. Subtract p from both sides:

   2q = 6 - p

3. Divide both sides by 2:

   q = (6 - p)/2.