Mathematical Practices 2 Reason abstractly and quantitatively 6

Mathematical Practices 2 Reason abstractly and quantitatively. 6 Attend to precision.

You graphed and solved linear equations. • Solve systems of linear equations graphically. • Solve systems of linear equations algebraically.

• break-even point • independent • system of equations • dependent • consistent • substitution method • inconsistent • elimination method

Solve by Using a Table Solve the system of equations by completing a table. x+y=3 – 2 x + y = – 6 Solve for y in each equation. x+y = 3 y = –x + 3 – 2 x + y = – 6 y = 2 x – 6

Solve by Using a Table Use a table to find the solution that satisfies both equations. Answer:

Solve by Using a Table Use a table to find the solution that satisfies both equations. Answer: The solution to the system is (3, 0).

What is the solution of the system of equations? x+y=2 x – 3 y = – 6 A. (1, 1) B. (0, 2) C. (2, 0) D. (– 4, 6)

Solve by Graphing Solve the system of equations by graphing. x – 2 y = 0 x+y=6 Write each equation in slope-intercept form. The graphs appear to intersect at (4, 2).

Solve by Graphing Check Substitute the coordinates into each equation. x – 2 y = 0 ? x+y =6 ? 4 – 2(2) = 0 4+2 =6 0=0 6=6 Answer: Original equations Replace x with 4 and y with 2. Simplify.

Solve by Graphing Check Substitute the coordinates into each equation. x – 2 y = 0 ? x+y =6 ? 4 – 2(2) = 0 4+2 =6 0=0 6=6 Original equations Replace x with 4 and y with 2. Simplify. Answer: The solution of the system is (4, 2).

Which graph shows the solution to the system of equations below? x + 3 y = 7 x–y = 3 A. C. B. D.

Classify Systems A. Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. x–y=5 x + 2 y = – 4 Write each equation in slope-intercept form.

Classify Systems Answer:

Classify Systems Answer: The graphs of the equations intersect at (2, – 3). Since there is one solution to this system, this system is consistent and independent.

Classify Systems B. Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. 9 x – 6 y = – 6 6 x – 4 y = – 4 Write each equation in slope-intercept form. Since the equations are equivalent, their graphs are the same line.

Classify Systems Answer:

Classify Systems Answer: Any ordered pair representing a point on that line will satisfy both equations. So, there are infinitely many solutions. This system is consistent and dependent.

Classify Systems C. Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. 15 x – 6 y = 0 5 x – 2 y = 10 Write each equation in slope-intercept form.

Classify Systems Answer:

Classify Systems Answer: The lines do not intersect. Their graphs are parallel lines. So, there are no solutions that satisfy both equations. This system is inconsistent.

Classify Systems D. Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. f(x) = – 0. 5 x + 2 g(x) = – 0. 5 x + 2 h(x) = 0. 5 x + 2

Classify Systems Answer:

Classify Systems Answer: f(x) and g(x) are consistent and dependent. f(x) and h(x) are consistent and independent. g(x) and h(x) are consistent and independent.

A. Graph the system of equations below. What type of system of equations is shown? x+y=5 2 x = y – 5 A. consistent and independent B. consistent and dependent C. consistent D. none of the above

B. Graph the system of equations below. What type of system of equations is shown? x+y=3 2 x = – 2 y + 6 A. consistent and independent B. consistent and dependent C. inconsistent D. none of the above

C. Graph the system of equations below. What type of system of equations is shown? y = 3 x + 2 – 6 x + 2 y = 10 A. consistent and independent B. consistent and dependent C. inconsistent D. none of the above

D. Graph the system of equations below. Which statement is not true? f(x) = x + 2 g(x) = x + 4 A. f(x) and g(x) are consistent and dependent. B. f(x) and g(x) are inconsistent. C. f(x) and h(x) are consistent and independent. D. g(x) and h(x) are consistent.

Use the Substitution Method FURNITURE Lancaster Woodworkers Furniture Store builds two types of wooden outdoor chairs. A rocking chair sells for $265 and an Adirondack chair with footstool sells for $320. The books show that last month, the business earned $13, 930 for the 48 outdoor chairs sold. How many of each chair were sold? Understand You are asked to find the number of each type of chair sold.

Use the Substitution Method Plan Define variables and write the system of equations. Let x represent the number of rocking chairs sold and y represent the number of Adirondack chairs sold. x + y = 48 265 x + 320 y = 13, 930 The total number of chairs sold was 48. The total amount earned was $13, 930.

Use the Substitution Method Solve one of the equations for one of the variables in terms of the other. Since the coefficient of x is 1, solve the first equation for x in terms of y. x + y = 48 x = 48 – y First equation Subtract y from each side.

Use the Substitution Method Solve Substitute 48 – y for x in the second equation. 265 x + 320 y = 13, 930 265(48 – y) + 320 y = 13, 930 12, 720 – 265 y + 320 y = 13, 930 55 y = 1210 y = 22 Second equation Substitute 48 – y for x. Distributive Property Simplify. Divide each side by 55.

Use the Substitution Method Now find the value of x. Substitute the value for y into either equation. x + y = 48 x + 22 = 48 x = 26 Answer: First equation Replace y with 22. Subtract 22 from each side.

Use the Substitution Method Now find the value of x. Substitute the value for y into either equation. x + y = 48 x + 22 = 48 x = 26 First equation Replace y with 22. Subtract 22 from each side. Answer: They sold 26 rocking chairs and 22 Adirondack chairs.

Use the Substitution Method Check You can use a graphing calculator to check this solution.

AMUSEMENT PARKS At Amy’s Amusement Park, tickets sell for $24. 50 for adults and $16. 50 for children. On Sunday, the amusement park made $6405 from selling 330 tickets. How many of each kind of ticket was sold? A. 210 adult; 120 children B. 120 adult; 210 children C. 300 children; 30 adult D. 300 children; 30 adult

Solve by Using Elimination Use the elimination method to solve the system of equations. x + 2 y = 10 x+y=6 In each equation, the coefficient of x is 1. If one equation is subtracted from the other, the variable x will be eliminated. x + 2 y = 10 (–)x + y = 6 y= 4 Subtract the equations.

Solve by Using Elimination Now find x by substituting 4 for y in either original equation. x+y =6 Second equation x+4 =6 Replace y with 4. x =2 Answer: Subtract 4 from each side.

Solve by Using Elimination Now find x by substituting 4 for y in either original equation. x+y =6 Second equation x+4 =6 Replace y with 4. x =2 Subtract 4 from each side. Answer: The solution is (2, 4).

Use the elimination method to solve the system of equations. What is the solution to the system? x + 3 y = 5 x + 5 y = – 3 A. (2, – 1) B. (17, – 4) C. (2, 1) D. no solution

No Solution and Infinite Solutions Solve the system of equations. 2 x + 3 y = 12 5 x – 2 y = 11 A. (2, 3) C. (0, 5. 5) B. (6, 0) D. (3, 2) Read the Test Item You are given a system of two linear equations and are asked to find the solution.

No Solution and Infinite Solutions Solve the Test Item Multiply the first equation by 2 and the second equation by 3. Then add the equations to eliminate the y variable. 2 x + 3 y = 12 Multiply by 2. 5 x – 2 y = 11 Multiply by 3. 4 x + 6 y = 24 (+)15 x – 6 y = 33 19 x = 57 x =3

No Solution and Infinite Solutions Replace x with 3 and solve for y. 2 x + 3 y = 12 2(3) + 3 y = 12 6 + 3 y = 12 3 y = 6 y=2 Answer: First equation Replace x with 3. Multiply. Subtract 6 from each side. Divide each side by 3.

No Solution and Infinite Solutions Replace x with 3 and solve for y. 2 x + 3 y = 12 2(3) + 3 y = 12 6 + 3 y = 12 3 y = 6 y=2 First equation Replace x with 3. Multiply. Subtract 6 from each side. Divide each side by 3. Answer: The solution is (3, 2). The correct answer is D.

Solve the system of equations. x + 3 y = 7 2 x + 5 y = 10 A. B. (1, 2) C. (– 5, 4) D. no solution