Systems of Equations and Methods of Solution (Chapter 5 Review)

Systems of Equations and Methods of Solution

Graphing Linear Equations

Graphing is a fundamental method for visualizing solutions to linear equations. The slope-intercept form, $y = mx + b$, is commonly used to plot straight lines.

• Slope ($m$): Indicates the steepness and direction of the line.

• Y-intercept ($b$): The point where the line crosses the y-axis.

• Example: For $y = \frac{1}{2}x + 2$, the slope is $\frac{1}{2}$ and the y-intercept is $2$.

To sketch:

• Plot the y-intercept $(0, 2)$.

• From this point, use the slope to find another point: rise 1, run 2.

• Draw the line through these points.

Solving Systems of Linear Equations

Systems of equations can be solved using several methods, each with its own advantages depending on the system's structure.

1. Graphical Solution

• Plot both equations on the same axes.

• The intersection point(s) represent the solution(s).

• Example: Solve $x + 2y = 3$ and $2x - y = 4$ graphically.

2. Substitution Method

• Solve one equation for one variable.

• Substitute this expression into the other equation.

• Solve for the remaining variable, then back-substitute.

• Example:

  • Given $x + 2y = 3$ and $2x - y = 4$.

  • Solve $x = 3 - 2y$.

  • Substitute into the second: $2(3 - 2y) - y = 4$.

  • Solve for $y$, then $x$.

3. Elimination (Addition/Subtraction) Method

• Multiply equations if necessary to align coefficients.

• Add or subtract equations to eliminate one variable.

• Solve for the remaining variable, then back-substitute.

• Example:

  • Given $3x - 2y = 5$ and $2x + y = 3$.

  • Multiply the second equation by 2: $4x + 2y = 6$.

  • Add to the first: $(3x - 2y) + (4x + 2y) = 5 + 6$.

  • Solve for $x$, then $y$.

4. Determinants (Cramer's Rule)

• For a system:

  • $a_1x + b_1y = c_1$

  • $a_2x + b_2y = c_2$

• Determinant $D = \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix} = a_1b_2 - a_2b_1$

• $x = \frac{\begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix}}{D}$

• $y = \frac{\begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix}}{D}$

• Example: Solve $7x + 2y = 6$, $2x - y = -1$ using determinants.

Word Problems: Mixture Problems

Mixture problems involve combining two or more substances with different properties to achieve a desired result.

• Example: Two types of copper ore are mixed to obtain a specific amount and concentration.

• Let $x$ = tonnes of 6.0% copper ore, $y$ = tonnes of 4.2% copper ore.

• Set up equations:

  • $x + y = 42$ (total mass)

  • $0.06x + 0.042y = 2$ (total copper content)

• Solve the system using substitution or elimination.

• Solution: $x = 27.6$ tonnes, $y = 14.4$ tonnes.

Summary Table: Methods for Solving Systems of Equations

Method	Steps	Best Use
Graphical	Plot both equations, find intersection	Visual understanding, simple systems
Substitution	Solve for one variable, substitute	One equation easily solved for a variable
Elimination	Add/subtract to eliminate variable	Coefficients easily aligned
Determinants	Apply Cramer's Rule	Exact solutions for 2x2 systems

Additional info: The notes and questions cover key Precalculus topics in systems of equations, including graphical, algebraic, and determinant methods, as well as real-world applications such as mixture problems.