Download GRAPHING LINEAR EQUATIONS IN TWO VARIABLES

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Partial differential equation wikipedia, lookup

Exact solutions in general relativity wikipedia, lookup

Differential equation wikipedia, lookup

Schwarzschild geodesics wikipedia, lookup

Transcript
GRAPHING LINEAR EQUATIONS IN TWO VARIABLES
To graph a linear inequality with one variable we use the number line. To graph a linear inequality
with two variables we use a two-dimensional graph. As with linear equations, linear inequalities can be
written in standard form or slope intercept form.
Standard Form:
Ax + By < C or Ax + By > C
Ax + By ≤ C or Ax + By ≥ C
y < mx + b or y > mx + b
y ≤ mx + b or y ≥ mx + b
Slope-Intercept Form:
To graph a linear inequality with two variables it is probably easier and recommended to change it to
slope intercept form. There are two important rules to remember when graphing a linear inequality. If
the inequality sign is > or <, the line is dashed. If the inequality is ≥ or ≤ , then the line is solid. Also,
for inequalities that are greater than > or greater or equal to ≥ the graph needs to be shaded above the
line. For inequalities that are less than < or less or equal to ≤ the graph needs to be shaded below the
line.
Examples:
Graph the solutions of 3 x + 2 y ≥ 6 :
Step 1 – Since the equation is in standard form, change it to slope intercept form.
3x + 2 y ≥ 6
→
2 y ≥ −3 x + 6
−3 x + 6
y≥
2
3
y ≥ − x+3
2
← Slope-Intercept Form
Step 2 – Graph the slope-intercept form equation. Since the inequality has the
sign ≥ the line is solid.
Step 3 – Since the equation has the sign ≥ , the graph is shaded above the line.
3
← The shaded area in the
graph is the graphic
answer to the inequality.
It means that any point
above the line, including
the line, satisfies the
inequality and can be a
possible answer.
Graph the solutions of x − 3 y < 4
Step 1 – Since the equation is in standard form, change it to slope intercept form.
x − 3y < 4
→
− 3y < −x + 4
−x + 4
y>
−3
Remember to switch the
inequality sign when
dividing by a negative.
y>
1
4
x−
3
3
← Slope-Intercept Form
Step 2 – Graph the slope-intercept form equation. Since the inequality has the
sign > the line is dashed.
Step 3 – Since the equation has the sign >, the graph is shaded above the line.
4
Solving a System of Linear Inequalities
To solve a system of linear inequalities we need to do it graphically since the solution to a system of
linear inequalities is the set of points whose coordinates satisfy all the inequalities in the system, or
where both shaded areas overlap. Graph the lines as previously shown.
1
y <− x+2
Example:
Determine the solution to the following system of inequalities
2
x− y ≤4
Step 1 – Change the second inequality to slope-intercept form.
x− y≤4
x− x− y ≤ 4− x
− y ≤ −x + 4
y ≥ x−4
← Remember to switch the inequality sign since
we are dividing by a negative number.
Step 2 – Graph both inequalities in slope-intercept form and shade the solution
for each inequality. Since the first inequality has the sign < the line is
dashed. The second inequality has the sign ≥ so the line is solid.
y ≥ x−4
Solution to the
system of
inequalities
y<−
5
1
x+2
2
GRAPHING LINEAR INEQUALITIES IN TWO VARIABLES – EXERCISES
Graph each inequality:
1
1. y ≥ − x
2
2. y < 2 x + 1
3. 10 ≥ 5 x − 2 y
Determine the solution to each system of inequalities:
2x − y < 4
y ≥ −x + 2
5.
y < 3x − 4
6x ≥ 2 y + 8
x≥0
1
1
7. x + y ≥ 2
2
2
2 x − 3 y ≤ −6
x≥0
y≥0
8.
2x + 3y ≤ 6
4x + y ≤ 4
x≥0
y≥0
9.
5 x + 4 y ≤ 16
x + 6 y ≤ 18
4.
6.
− 3 x + 2 y ≥ −5
y ≤ −4 x + 7
x≥0
y≥0
10. x ≤ 15
30 x + 25 y ≤ 750
10 x + 40 y ≤ 800
6
GRAPHING LINEAR INEQUALITIES IN TWO VARIABLES – ANSWERS TO EXERCISES
1.
2.
3.
4.
5.
6.
7.
8.
7
9.
10.
8