Grade 9 Solving Linear Equations in Two Variables

System of Linear Equations: A set of two or more linear equations involving the same variables. In this chapter, we focus on systems of two linear equations in two variables (like x and y).

Solution of a System: An ordered pair (x, y) that satisfies **all** the equations in the system simultaneously. When graphed, this solution is the point where the lines of the equations intersect.

Types of Systems:

• **Consistent System:** Has at least one solution (lines intersect or are the same).
• **Inconsistent System:** Has no solution (lines are parallel and distinct).

Unique Solution: A consistent system where the lines intersect at exactly one point has a unique solution.

Step 1: The elimination method involves eliminating one variable by adding or subtracting the equations. Notice that the coefficients of 'y' are already opposites (+1 and -1).

Step 2: Add Equation 1 and Equation 2.
(2x + y) + (x - y) = 5 + 1
2x + y + x - y = 6

Step 3: Combine like terms and solve for x. The 'y' terms cancel out.
(2x + x) + (y - y) = 6
3x + 0 = 6
3x = 6
x = 6 / 3
x = 2

Step 4: Substitute the value of x (which is 2) into either of the original equations to solve for y. Let's use Equation 1: 2x + y = 5.
2(2) + y = 5
4 + y = 5

Step 5: Solve for y.
y = 5 - 4
y = 1

Step 6: The solution is the ordered pair (x, y) = (2, 1).

Step 1: The substitution method involves solving one equation for one variable, and then substituting that expression into the other equation. From Equation 1 (x + 2y = 4), it's easy to solve for x:
x = 4 - 2y (Eq 3)

Step 2: Substitute the expression for x from Equation 3 into Equation 2 (3x - y = 5).
3(4 - 2y) - y = 5

Step 3: Solve the resulting equation for y.
12 - 6y - y = 5
12 - 7y = 5
-7y = 5 - 12
-7y = -7
y = -7 / -7
y = 1

Step 4: Substitute the value of y (which is 1) back into Equation 3 (x = 4 - 2y) to solve for x.
x = 4 - 2(1)
x = 4 - 2
x = 2

Step 5: The solution is the ordered pair (x, y) = (2, 1).

Step 1: To use elimination, we need the coefficients of one variable to be the same or opposites. We can multiply Equation 2 by 2 to make the coefficient of 'y' 2, matching Equation 1.
Multiply Equation 2 by 2: 2(2x + y) = 2(4)
4x + 2y = 8 (Eq 3)

Step 2: Now we have the system:
3x + 2y = 7 (Eq 1)
4x + 2y = 8 (Eq 3) The coefficients of 'y' are the same (2). Subtract Equation 1 from Equation 3 to eliminate 'y'.
(4x + 2y) - (3x + 2y) = 8 - 7
4x + 2y - 3x - 2y = 1

Step 3: Combine like terms and solve for x.
(4x - 3x) + (2y - 2y) = 1
x + 0 = 1
x = 1

Step 4: Substitute the value of x (which is 1) into either original equation to solve for y. Using Equation 2: 2x + y = 4.
2(1) + y = 4
2 + y = 4

Step 5: Solve for y.
y = 4 - 2
y = 2

Step 6: The solution is (x, y) = (1, 2).