Find the number of solutions to a system of equations

Learning Objectives Define what a system of linear equations is and what its solution represents. Identify the three possible outcomes for the number of solutions to a system of linear equations (one, none, or infinitely many). Graphically determine the number of solutions to a system of linear equations by analyzing the intersection of lines. Algebraically determine the number of solutions by comparing the slopes and y-intercepts of the equations. Convert linear equations into slope-intercept form to facilitate comparison. Apply their understanding of the number of solutions to solve simple real-world problems. Have you ever tried to find the perfect meeting spot between two friends coming from different locations? 📍 That's a bit like finding the solution to a system...

TermDefinitionExample System of Linear EquationsA set of two or more linear equations that are considered together. For Grade 8, we usually focus on systems with two equations and two variables (x and y).Equation 1: $y = 2x + 1$ Equation 2: $y = -x + 4$ Solution to a SystemA pair of values (x, y) that makes ALL equations in the system true. Graphically, it's the point where the lines intersect.For the system $y = x + 1$ and $y = 2x - 1$, the point $(2, 3)$ is a solution because $3 = 2 + 1$ and $3 = 2(2) - 1$ are both true. Slope-Intercept FormA way to write linear equations as $y = mx + b$, where 'm' is the slope of the line and 'b' is the y-intercept (the point where the line crosses the y-axis).In $y = 3x - 2$, the slope is 3 and the y-intercept is -2. One Solut...

Rule for One Solution If the slopes of the two linear equations are different, then the system has exactly one solution. When you have two equations in slope-intercept form ($y = m_1x + b_1$ and $y = m_2x + b_2$), if $m_1 \neq m_2$, the lines will cross at one unique point. Rule for No Solution If the slopes of the two linear equations are the same, but their y-intercepts are different, then the system has no solution. For $y = m_1x + b_1$ and $y = m_2x + b_2$, if $m_1 = m_2$ and $b_1 \neq b_2$, the lines are parallel and will never intersect. Rule for Infinitely Many Solutions If the slopes of the two linear equations are the same, AND their y-intercepts are also the same, then the system has infinitely many solutions. For $y = m_1x + b_1$ and $y = m_2x + b_2$, if $...