Solving Systems of Linear Equations: Fractions, Elimination, and Substitution Methods 3.1

Solving Systems of Linear Equations

Introduction

Systems of linear equations are sets of two or more linear equations involving the same variables. In College Algebra, solving such systems is a fundamental skill, often using methods such as substitution, elimination, and handling equations with fractional coefficients.

Solving Systems with Fractional Coefficients

Equations with fractional coefficients can be simplified by clearing the fractions, making the system easier to solve.

• Key Point 1: Multiply both sides of each equation by the least common denominator (LCD) to eliminate fractions.

• Key Point 2: After clearing fractions, use substitution or elimination to solve the system.

• Example: Given the system: $3x-y=6$ $\frac{2}{3}x - \frac{1}{2}y = \frac{7}{6}$

• Step 1: Solve the first equation for $y$: $y=3x-6$

• Step 2: Substitute $y$ into the second equation: $\frac{2}{3}x-\frac{1}{2}(3x-6)=\frac{7}{6}$

• Step 3: Multiply both sides by 6 (LCD): $4x-3(3x-6)=7$ $4x-9x+18=7$ $-5x=-11$ $x=\frac{11}{5}$

• Step 4: Substitute $x$ back to find $y$: $y=3(\frac{11}{5})-6\to y=\frac{33}{5}-\frac{30}{5}=\frac{33-30}{5}\to y=\frac35$

• Solution: $(x,y)=(\frac{11}{5},-\frac35)$

Step-by-Step Procedure for Solving Systems (Substitution Method)

• 1. Solve for fractions in equation 2.

• 2. Multiply by the LCD to clear fractions.

• 3. Substitute the expression from one equation into the other.

• 4. Solve for one variable.

• 5. Substitute back to find the other variable.

• 6. Write the solution as an ordered pair $(x, y)$.

• 7. Check the solution in both original equations.

Solving Systems Using the Elimination Method

The elimination method involves adding or subtracting equations to eliminate one variable, making it possible to solve for the other.

• Key Point 1: Arrange both equations in the form $Ax + By = C$.

• Key Point 2: Multiply one or both equations by appropriate numbers so that the coefficients of one variable are opposites.

• Key Point 3: Add or subtract the equations to eliminate one variable.

• Key Point 4: Solve for the remaining variable, then substitute back to find the other.

• Example: $2x + 3y = 6$ $4x-3y=6$

• Step 1: Add the first from the second: $(2x+3y=-6)+(4x-3y)=6$ $6x=0\implies x=0$

• Substitute $x = 0$ into the first equation: $2(0)+3y=6\implies3y=-6\implies y=-2$ Solution: $(0,-2)$

Step-by-Step Procedure for Elimination

• 1. Add or subtract equations to eliminate one variable.

• 2. Solve for the remaining variable.

• 3. Substitute back to find the other variable.

• 4. Write the solution as an ordered pair $(x, y)$.

• 5. (Optional) Convert equations to slope-intercept form for graphing.

General Tips and Additional Information

• When using decimals, multiply each term by 10, 100, 1000, etc., to clear decimals.

• Always check your solution by substituting both values into both original equations.

• Write both equations in standard form ($Ax + By = C$) before using elimination.

• Multiplying both equations by appropriate values can help align coefficients for elimination.

Multiplying to Find Least Common Denominator (LCD)

• Key Point: To clear fractions, multiply each term by the LCD of all denominators in the equation.

• Example: For denominators 2, 3, and 5, the LCD is $2 \times 3 \times 5 = 30$.

Summary Table: Methods for Solving Systems of Equations

Method	When to Use	Steps
Substitution	When one equation is easily solved for one variable	1. Solve for one variable 2. Substitute into other equation 3. Solve for second variable 4. Back-substitute
Elimination	When both equations are in standard form	1. Align coefficients 2. Add/subtract equations 3. Solve for one variable 4. Back-substitute
Graphing	For visual solutions or checking	1. Convert to slope-intercept form 2. Graph both lines 3. Intersection is the solution

Additional info:

• Some steps and examples have been expanded for clarity and completeness.

• Checking solutions in both equations ensures accuracy and helps avoid arithmetic errors.