Review the key concepts, formulae, and examples before starting your quiz.
🔑Concepts
A system of linear equations consists of two or more equations with the same set of variables. Solving the system simultaneously means finding the unique values for the variables that satisfy all equations at the same time.
The Graphical Solution represents each equation as a straight line on a coordinate plane. The solution to the system is the point where the two lines intersect. Visually, if you plot two lines and they cross at a single point, that specific coordinate is the only solution.
The Substitution Method involves isolating one variable in one equation (e.g., expressing in terms of ) and substituting this expression into the second equation. This reduces the system to a single linear equation with one variable, which can then be solved using standard algebraic steps.
The Elimination Method (or Addition Method) is used to remove one variable by adding or subtracting the equations. To do this, you may need to multiply one or both equations by a constant so that the coefficients of one variable are identical or opposites. Visually, this is equivalent to finding where the lines meet by manipulating their algebraic forms.
Parallel Lines and No Solution: If two lines have the same slope () but different -intercepts (), they are parallel and will never intersect. Visually, these lines run side-by-side forever. Algebraically, this results in an impossible statement like .
Coincident Lines and Infinite Solutions: If two equations represent the same line (same slope and same -intercept), they lie exactly on top of each other. Every point on the line is a solution. Algebraically, this results in a true identity such as or .
Translating Word Problems: Many real-world scenarios (like comparing two different rental costs) can be modeled as systems. Identify the two unknowns, assign them variables like and , and create two distinct equations based on the constraints provided in the problem.
📐Formulae
Slope-Intercept Form:
Standard Form:
A general system of equations:
Slope Formula:
💡Examples
Problem 1:
Solve the following system using the substitution method:
Solution:
Step 1: Since the first equation is already solved for , substitute into the second equation for : Step 2: Simplify and solve for : Step 3: Substitute back into the first equation to find : Step 4: State the final solution as a coordinate pair: .
Explanation:
The substitution method is most efficient here because one variable is already isolated. By replacing in the second equation, we create a linear equation in one variable (), solve it, and then back-substitute to find the other coordinate.
Problem 2:
Solve the following system using the elimination method:
Solution:
Step 1: Observe that the coefficients of are opposites ( and ). Add the two equations together to eliminate : Step 2: Solve for : Step 3: Substitute into the first equation to solve for : Step 4: The solution is , or .
Explanation:
The elimination method is ideal here because the terms cancel out immediately upon addition. This method is often the fastest way to solve systems in standard form .