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

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

Which graph shows the solution to the system of equations below? x + 3y = 7 x – y = 3 D. Example 2

Write each equation in slope-intercept form. Classify Systems A. Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. x – y = 5 x + 2y = –4 Write each equation in slope-intercept form. Example 3

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. Example 3

Write each equation in slope-intercept form. Classify Systems B. Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. 9x – 6y = –6 6x – 4y = –4 Write each equation in slope-intercept form. Since the equations are equivalent, their graphs are the same line. Example 3

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. Example 3

Write each equation in slope-intercept form. Classify Systems C. Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. 15x – 6y = 0 5x – 2y = 10 Write each equation in slope-intercept form. Example 3

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. Example 3

Classify Systems D. Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. f(x) = –0.5x + 2 g(x) = –0.5x + 2 h(x) = 0.5x + 2 Example 3

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. Example 3

A. consistent and independent B. consistent and dependent A. Graph the system of equations below. What type of system of equations is shown? x + y = 5 2x = y – 5 A. consistent and independent B. consistent and dependent C. consistent D. none of the above Example 3

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

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

Concept

Concept

You are asked to find the number of each type of chair sold. 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. Example 4

x + y = 48 The total number of chairs sold was 48. 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 The total number of chairs sold was 48. 265x + 320y = 13,930 The total amount earned was $13,930. Example 4

x = 48 – y Subtract y from each side. 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 First equation x = 48 – y Subtract y from each side. Example 4

Solve Substitute 48 – y for x in the second equation. Use the Substitution Method Solve Substitute 48 – y for x in the second equation. 265x + 320y = 13,930 Second equation 265(48 – y) + 320y = 13,930 Substitute 48 – y for x. 12,720 – 265y + 320y = 13,930 Distributive Property 55y = 1210 Simplify. y = 22 Divide each side by 55. Example 4

x = 26 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 First equation x + 22 = 48 Replace y with 22. x = 26 Subtract 22 from each side. Answer: They sold 26 rocking chairs and 22 Adirondack chairs. Example 4

Check You can use a graphing calculator to check this solution. Use the Substitution Method Check You can use a graphing calculator to check this solution. Example 4

AMUSEMENT PARKS At Amy’s Amusement Park, tickets sell for $24 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 Example 4

Concept

Use the elimination method to solve the system of equations. Solve by Using Elimination Use the elimination method to solve the system of equations. x + 2y = 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 + 2y = 10 (–)x + y = 6 y = 4 Subtract the equations. Example 5

Now find x by substituting 4 for y in either original equation. 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). Example 5

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

Solve the system of equations. 2x + 3y = 12 5x – 2y = 11 No Solution and Infinite Solutions Solve the system of equations. 2x + 3y = 12 5x – 2y = 11 A. (2, 3) B. (6, 0) C. (0, 5.5) D. (3, 2) Read the Test Item You are given a system of two linear equations and are asked to find the solution. Example 6

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. 2x + 3y = 12 4x + 6y = 24 Multiply by 2. Multiply by 3. 5x – 2y = 11 (+)15x – 6y = 33 19x = 57 x = 3 Example 6

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

Concept