warm up evaluate each expression for x = 1 and y =–3. 1. x – 4 y 2. –2 x + y
DESCRIPTION
Warm Up Evaluate each expression for x = 1 and y =–3. 1. x – 4 y 2. –2 x + y Write each expression in slope-intercept form. 3. y – x = 1 4. 2 x + 3 y = 6 5. 0 = 5 y + 5 x. 13. –5. y = x + 1. y = x + 2. y = – x. Objectives. - PowerPoint PPT PresentationTRANSCRIPT
Holt Algebra 1
6-1 Solving Systems by Graphing
Warm UpEvaluate each expression for x = 1 and y =–3.
1. x – 4y 2. –2x + y
Write each expression in slope-
intercept form.
3. y – x = 1
4. 2x + 3y = 6
5. 0 = 5y + 5x
13 –5
y = x + 1
y = x + 2
y = –x
Holt Algebra 1
6-1 Solving Systems by Graphing
Identify solutions of linear equations in two variables.
Solve systems of linear equations in two variables by graphing.
Objectives
Holt Algebra 1
6-1 Solving Systems by Graphing
systems of linear equationssolution of a system of linear equations
Vocabulary
Holt Algebra 1
6-1 Solving Systems by Graphing
A system of linear equations is a set of two or more linear equations containing two or more variables.
A solution of a system of linear equations with two variables is an ordered pair that satisfies each equation in the system. So, if an ordered pair is a solution, it will make both equations true.
Holt Algebra 1
6-1 Solving Systems by Graphing
Tell whether the ordered pair is a solution of the given system.
Example 1A: Identifying Systems of Solutions
(5, 2);
The ordered pair (5, 2) makes both equations true.(5, 2) is the solution of the system.
Substitute 5 for x and 2 for y in each equation in the system.
3x – y = 13
2 – 2 00 0
0 3(5) – 2 13
15 – 2 13
13 13
3x – y 13
Holt Algebra 1
6-1 Solving Systems by Graphing
If an ordered pair does not satisfy the first equation in the system, there is no reason to check the other equations.
Helpful Hint
Holt Algebra 1
6-1 Solving Systems by Graphing
Example 1B: Identifying Systems of Solutions
Tell whether the ordered pair is a solution of the given system.
(–2, 2);x + 3y = 4–x + y = 2
–2 + 3(2) 4
x + 3y = 4
–2 + 6 44 4
–x + y = 2
–(–2) + 2 24 2
Substitute –2 for x and 2 for y in each equation in the system.
The ordered pair (–2, 2) makes one equation true but not the other.
(–2, 2) is not a solution of the system.
Holt Algebra 1
6-1 Solving Systems by Graphing
Check It Out! Example 1a
Tell whether the ordered pair is a solution of the given system.
(1, 3); 2x + y = 5–2x + y = 1
2x + y = 5
2(1) + 3 52 + 3 5
5 5
The ordered pair (1, 3) makes both equations true.
Substitute 1 for x and 3 for y in each equation in the system.
–2x + y = 1
–2(1) + 3 1–2 + 3 1
1 1
(1, 3) is the solution of the system.
Holt Algebra 1
6-1 Solving Systems by Graphing
Check It Out! Example 1b
Tell whether the ordered pair is a solution of the given system.
(2, –1); x – 2y = 43x + y = 6
The ordered pair (2, –1) makes one equation true, but not the other.
Substitute 2 for x and –1 for y in each equation in the system.
(2, –1) is not a solution of the system.
3x + y = 6
3(2) + (–1) 66 – 1 6
5 6
x – 2y = 4
2 – 2(–1) 42 + 2 4
4 4
Holt Algebra 1
6-1 Solving Systems by Graphing
All solutions of a linear equation are on its graph. To find a solution of a system of linear equations, you need a point that each line has in common. In other words, you need their point of intersection.
y = 2x – 1
y = –x + 5
The point (2, 3) is where the two lines intersect and is a solution of both equations, so (2, 3) is the solution of the systems.
Holt Algebra 1
6-1 Solving Systems by Graphing
Sometimes it is difficult to tell exactly where the lines cross when you solve by graphing. It is good to confirm your answer by substituting it into both equations.
Helpful Hint
Holt Algebra 1
6-1 Solving Systems by Graphing
Solve the system by graphing. Check your answer.Example 2A: Solving a System Equations by Graphing
y = xy = –2x – 3 Graph the system.
The solution appears to be at (–1, –1).
(–1, –1) is the solution of the system.
CheckSubstitute (–1, –1) into the system.
y = x
y = –2x – 3
• (–1, –1)
y = x
(–1) (–1)
–1 –1
y = –2x – 3
(–1) –2(–1) –3
–1 2 – 3–1 – 1
Holt Algebra 1
6-1 Solving Systems by Graphing
Solve the system by graphing. Check your answer.Example 2B: Solving a System Equations by Graphing
y = x – 6
Rewrite the second equation in slope-intercept form.
y + x = –1Graph using a calculator and then use the intercept command.
y = x – 6
y + x = –1
− x − x
y =
Holt Algebra 1
6-1 Solving Systems by Graphing
Solve the system by graphing. Check your answer.Example 2B Continued
Check Substitute into the system.
y = x – 6
The solution is .
+ – 1
–1
–1
–1 – 1
y = x – 6
– 6
Holt Algebra 1
6-1 Solving Systems by Graphing
Solve the system by graphing. Check your answer.Check It Out! Example 2a
y = –2x – 1 y = x + 5 Graph the system.
The solution appears to be (–2, 3).
Check Substitute (–2, 3) into the system.
y = x + 5
3 –2 + 5
3 3
y = –2x – 1
3 –2(–2) – 1
3 4 – 1
3 3(–2, 3) is the solution of the system.
y = x + 5
y = –2x – 1
Holt Algebra 1
6-1 Solving Systems by Graphing
Solve the system by graphing. Check your answer.Check It Out! Example 2b
2x + y = 4
Rewrite the second equation in slope-intercept form.
2x + y = 4–2x – 2x
y = –2x + 4
Graph using a calculator and then use the intercept command.
2x + y = 4
Holt Algebra 1
6-1 Solving Systems by Graphing
Solve the system by graphing. Check your answer.Check It Out! Example 2b Continued
2x + y = 4
The solution is (3, –2).
Check Substitute (3, –2) into the system.
2x + y = 42(3) + (–2) 4
6 – 2 44 4
2x + y = 4
–2 (3) – 3
–2 1 – 3
–2 –2
Holt Algebra 1
6-1 Solving Systems by Graphing
Lesson Quiz: Part I
Tell whether the ordered pair is a solution of the given system.
1. (–3, 1);
2. (2, –4);
yes
no
Holt Algebra 1
6-1 Solving Systems by Graphing
Lesson Quiz: Part II
Solve the system by graphing.
3. (2, 5)y + 2x = 9
y = 4x – 3