Systems of Linear Equations
A system of linear equations is a collection of two or more linear equations involving the same set of variables. In Grade 8, you will primarily focus on systems of two linear equations and, occasionally, three linear equations.
Key Concepts
1. System of Two Linear Equations
A system of two linear equations with two variables \(x\) and \(y\) can be written as:
\[ \begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases} \]
Here, \(a_1\), \(b_1\), \(c_1\), \(a_2\), \(b_2\), and \(c_2\) are constants.
2. System of Three Linear Equations
A system of three linear equations with three variables \(x\), \(y\), and \(z\) can be written as:
\[ \begin{cases} a_1x + b_1y + c_1z = d_1 \\ a_2x + b_2y + c_2z = d_2 \\ a_3x + b_3y + c_3z = d_3 \end{cases} \]
Here, \(a_1\), \(b_1\), \(c_1\), \(d_1\), \(a_2\), \(b_2\), \(c_2\), \(d_2\), \(a_3\), \(b_3\), \(c_3\), and \(d_3\) are constants.
Solving Systems of Linear Equations
1. Graphical Method
In the graphical method, each equation in the system is graphed on the same coordinate plane. The solution to the system is the point where the lines intersect. For a system of two equations, this point is the solution \((x, y)\).
Example: Solve the system:
\[ \begin{cases} 2x + y = 4 \\ x - y = 1 \end{cases} \]
Graph both equations on the same coordinate plane. The lines intersect at the point \((2, 0)\), which is the solution.
2. Substitution Method
In the substitution method, solve one of the equations for one variable in terms of the other, and then substitute this expression into the other equation.
Example: Solve the system:
\[ \begin{cases} 2x + y = 4 \\ x - y = 1 \end{cases} \]
From the second equation, solve for \(x\): \(x = y + 1\).
Substitute \(x = y + 1\) into the first equation: \(2(y + 1) + y = 4\).
Solve for \(y\): \(2y + 2 + y = 4 \Rightarrow 3y + 2 = 4 \Rightarrow 3y = 2 \Rightarrow y = \frac{2}{3}\).
Substitute \(y = \frac{2}{3}\) back into \(x = y + 1\): \(x = \frac{2}{3} + 1 = \frac{5}{3}\).
The solution is \(\left(\frac{5}{3}, \frac{2}{3}\right)\).
3. Elimination Method
In the elimination method, multiply one or both equations by constants so that when the equations are added, one of the variables is eliminated.
Example: Solve the system:
\[ \begin{cases} 2x + y = 4 \\ x - y = 1 \end{cases} \]
Add the two equations: \(2x + y + x - y = 4 + 1 \Rightarrow 3x = 5 \Rightarrow x = \frac{5}{3}\).
Substitute \(x = \frac{5}{3}\) into the second equation: \(\frac{5}{3} - y = 1 \Rightarrow y = \frac{5}{3} - 1 = \frac{2}{3}\).
The solution is \(\left(\frac{5}{3}, \frac{2}{3}\right)\).
Analogies to Understand Systems of Linear Equations
1. Intersection of Paths
Think of each linear equation as a path on a map. The solution to the system is the point where these paths intersect. For example, if two friends are walking on different paths, the point where they meet is the solution to the system of their paths.
2. Balance of Forces
Consider a system of linear equations as a balance of forces. Each equation represents a force acting on a system. The solution to the system is the point where all forces are balanced, similar to how a seesaw balances when the forces on both sides are equal.