BackSolving Systems of Linear Equations: Methods and Applications
Study Guide - Smart Notes
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Section 9.1: Solving Systems of Linear Equations
Graphical Method
The graphical method involves plotting equations on a coordinate plane to find their intersection point(s), which represent the solution(s) to the system.
Step 1: Solve each equation for y. Plug into a graphing calculator for y and x.
Step 2: Find the point(s) of intersection on the graph (calculator or manual).
Step 3: If the lines are parallel, there is no solution (inconsistent system).
Step 4: If the graphs are the same line, there are infinitely many solutions (dependent system).
Example: Solve the system by graphing:
Substitution Method
The substitution method is useful when one equation can be easily solved for one variable. Substitute this expression into the other equation to solve for the remaining variable.
Step 1: Solve one equation for a variable.
Step 2: Substitute this expression into the other equation.
Step 3: Solve for the remaining variable.
Step 4: Substitute back to find the value of the first variable.
Example: Solve using the substitution method:
Elimination Method
The elimination method involves adding or subtracting equations to eliminate one variable, making it possible to solve for the other.
Step 1: Choose a variable to eliminate and find a common multiple of its coefficients.
Step 2: Multiply one or both equations as needed so the coefficients of the chosen variable are opposites.
Step 3: Add or subtract the equations to eliminate the variable.
Step 4: Solve for the remaining variable, then substitute back to find the other variable.
Example: Solve using the elimination method: Hint: Multiply each equation by 10 to clear decimals.
Special Cases in Systems of Equations
Sometimes, the substitution or elimination method reveals that the system has no solution (inconsistent) or infinitely many solutions (dependent). This occurs when the resulting equation is either a contradiction (e.g., ) or an identity (e.g., ).
Applications of Systems of Equations
Systems of equations are used to solve real-world problems involving multiple unknowns.
Investment Problems: Distribute a sum among accounts with different interest rates.
Mixture Problems: Blend items with different costs to achieve a target price.
Motion Problems: Find speeds of objects moving with or against a current.
Example:
You invested $8000 between two accounts paying 3% and 5% annual interest. If the total interest earned for the year was $340, how much was invested in each account?
The Coffee Counter charges $11 per pound for Kenyan French Roast and $10 per pound for Sumatran coffee. How much of each type should be used to make a 20-pound blend that sells for $10.35 per pound?
A riverboat travels 54 km downstream in 2 hours and 18 km upstream in 3 hours. Find the speed of the boat and the speed of the stream.
Key Formulas
General form of a linear equation:
System of two equations:
Comparison of Methods
Method | Best Use Case | Advantages | Disadvantages |
|---|---|---|---|
Graphical | Visualizing solutions, simple systems | Intuitive, quick for small systems | Imprecise for complex/decimal solutions |
Substitution | One equation easily solved for a variable | Algebraic, exact solutions | Can be cumbersome for complex systems |
Elimination | Both equations in standard form | Efficient for larger systems | Requires careful manipulation |
