Classify a system of equations

Learning Objectives Define a system of linear equations and its solution. Identify the slope and y-intercept of a linear equation. Classify a system of linear equations as having one solution, no solution, or infinitely many solutions by comparing their slopes and y-intercepts. Categorize systems of equations as consistent or inconsistent. Distinguish between independent and dependent systems of equations. Convert linear equations into slope-intercept form to facilitate classification. Explain the graphical interpretation of each classification type. Ever wonder if two different paths will ever cross, or if they'll always run parallel? 🛤️ In math, we can figure this out for lines too! In this lesson, you'll learn how to classify systems of linear equations based...

TermDefinitionExample System of Linear EquationsTwo or more linear equations working together. We are looking for values that satisfy ALL equations simultaneously.$\begin{cases} y = 2x + 1 \\ y = -x + 4 \end{cases}$ Solution to a SystemThe point(s) (x, y) that make ALL equations in the system true. Graphically, it's where the lines intersect.For the system $y = x + 1$ and $y = -x + 3$, the solution is $(1, 2)$ because $2 = 1 + 1$ and $2 = -1 + 3$ are both true. Slope-Intercept FormA way to write linear equations: $y = mx + b$, where 'm' is the slope and 'b' is the y-intercept.In $y = 3x - 5$, the slope (m) is 3 and the y-intercept (b) is -5. Consistent SystemA system of equations that has at least one solution (meaning the lines intersect at one point or are the s...

Classifying Systems: One Solution (Consistent & Independent) If the slopes of the two lines are different: $m_1 \neq m_2$ The lines will intersect at exactly one point, meaning there is one unique solution. The system is consistent and independent. Classifying Systems: No Solution (Inconsistent) If the slopes are the same, but the y-intercepts are different: $m_1 = m_2$ and $b_1 \neq b_2$ The lines are parallel and will never intersect, meaning there is no solution. The system is inconsistent. Classifying Systems: Infinitely Many Solutions (Consistent & Dependent) If both the slopes and the y-intercepts are the same: $m_1 = m_2$ and $b_1 = b_2$ The lines are identical (they are the same line), meaning every point on the line is a solution. There are infinitel...