Download 11 Systems of Equations and Inequalities

11
11.1
Systems of Equations and Inequalities
Systems of Linear Equations in Two Variables
Definition 11.1 A system of equations is a collection of two or more
equations, each containing one or more variables. A solution to a system
of equations consists of values for the variables that are solutions of each
equation of the system.
Steps for Solving by Substitution
• Pick one of the equations and solve for one of the variables in terms of
the remaining variables.
• Substitute into the OTHER equation.
• Solve this equation.
• Back-substitute.
1. Solve
2x + y = 1
3x + 4y = 14
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2. Solve
x2 + y 2 = 100
3x − y = 10
Elimination Method
• Adjust the coefficients by multiplying one or more of the equations
by appropriate numbers so that the coefficient of one variable in one
equation is the negative of its coefficient in the other equation.
• Add the equations to eliminate one variable, then solve for the remaining variable.
• Back-substitute and and solve for the remaining variable.
3. Solve
3x + 2y = 14
x − 2y = 2
2
4. Solve
x2 + 3y = 10
3x + 2y = 10
Number of solutions of a linear system in two variables
• The system has exactly one solution.
• The system has no solution.
• The system has infinitely many solutions.
5. Graph the linear system to determine if it has one solution, no solution,
or an infinite number of solutions. If there is exactly one solution, use
the graph to find it.
x + 2y = 4
2x + 4y = 8
3
6. Graph the linear system to determine if it has one solution, no solution,
or an infinite number of solutions. If there is exactly one solution, use
the graph to find it.
2x − y = 0
3x + 2y = 7
7. Solve the system or show that is has no solution. If the system has
infinitely many solutions, express them in ordered pair form.
4x + 5y = 10
10x + y = 11
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8. Solve the system or show that is has no solution. If the system has
infinitely many solutions, express them in ordered pair form.
2x + 5y = 1
−5x − 2y = 3
9. Solve the system or show that is has no solution. If the system has
infinitely many solutions, express them in ordered pair form.
3x − 5y = 3
5x + 5y = 21
10. Solve the system or show that is has no solution. If the system has
infinitely many solutions, express them in ordered pair form.
7x − y = 4
−21x + 3y = −12
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