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### High School Mathematics1.20 Linear Equations - Two Variables

 Linear equation with one variable: has one solution. Example: x + 4 = 12 x + 2x + 6 = 9 simplifies to 3x + 6 = 9 Linear equation with two variables: A linear equation in two variables has infinitely many solutions. Example: x + y = 12 For x = 1, y = 11, x = 2, y = 10, x = 3, y = 9, x = 4, y = 8..... and so on Linear equation with three variables Example: x + y + z = 12 Note: You will need one equation to solve for one variable, two equations to solve two variable and three equations to solve three variable and so on...... Method 1: Elimination Method: Solving Linear Equations with two variables. Solve for the values of a and b for the following equations: Example 1: a + b = 12 -----equation 1 a - b = 4 ------- equation 2. Adding both equations we get a + b = 12 a - b = 4 ------------------The terms +b and -b cancels, and we get 2a = 16 a = 16/8 = 2 a = 2 substituting the value of a in equation 1 we have a + b = 12 2 + b = 12 b = 12 - 2 b = 10 The solution of the two equations is a = 2 and b = 10 Example 2: c + 5b = 16 -----equation 1 c + 2b = 7 ------- equation 2 . Multiplying the equation 2 with (-1) we get c + 5b = 16 -----equation 1 -1(c + 2b = 7) => -c - 2b = -7-------equation 2 c + 5b = 16 -c - 2b = -7 ---------------------The terms c and -c cancel each other and we get 3b = 9 b = 9/3 = 3 substituting b in the equation c + 5b = 16 we get c + 5(3) = 16 c + 15 = 16 - 3 c = 16 - 15 c = 1 Example 3: 2a + 4b = 12 -----equation 1 a - b = 3 ------- equation 2 First pick a variable either a or b. Lets pick b. Second find the common multiple of b so we can eliminate that variable and find the value of the other variable. so multiply both sides of the equation 2 by 4 we have 2a + 4b = 12 -----equation 1 4(a - b = 3) => 4a - 4b = 12------- equation 2 Now adding equation 1 and 2 we have 2a + 4b = 12 4a - 4b = 12 ----------------------The terms +4b in equation 1 and -4b in equation 2 cancels, and we get 6a = 24 a = 24/6 =4 substituting value of a in equation a - b = 3 we get 4 - b = 3 4- 3 = b b = 1 Therefore a = 4 and b = 1 Verification: To verify substitute the values of a and b in the equations above 2a + 4b = 12 2a + 4b = 2(4) + 4(1) = 8 + 4 = 12 NOTE: In the above question if you pick b 2a + 4b = 12 -----equation 1 a - b = 3 ------- equation 2 multiplying equation 2 by 2 we have 2(a - b = 3) ------- equation 2 2a - 2b = 6 multiplying equation by -1 we gave -1(2a - 2b = 6) we get -2a + 2b = -6 Solving both equations we have 2a + 4b = 12 -----equation 1 -2a + 2b = -6-----equation 2 ------------------------- 6b = 6 b = 1 Example 4: 2a + b = 4 -----equation 1 3a + 2b = 3 ------- equation 2. Multiplying the equation 1 with 3 and equation 2 with (-2) we get 3(2a + b = 4) -----equation 1 -2(3a + 2b = 3) ------- equation 2 6a + 3b = 12 -----equation 1 -6a - 4b = -6) ------- equation 2 -----------------------The terms 6a and -6a cancel each other and we get -b = 6 b = -6 substituting b in the equation 2a + b = 4 we get 2a + (-6) = 4 2a - 6 = 4 2a = 4 + 6 2a = 10 a = 10/2 = 5 Method 2: Substitution Method: Solve the equations: 3x + 5y = 26 y = 2x substitute the second equation in the first 3x + 5(2x) = 26 3x + 10x = 26 13x = 26 x = 2 y = 2x = 2x2 = 4 Solve the equations: x + y = 26 ------- equation 1 x - y = 4 -------equation 2 x - y = 4 -------equation 2 can be written as x = y + 4. Substituting x in the equation 1 we have: x + y = 26 ------- equation 1 y + 4 + y = 26 2y + 4 = 26 2y + 4 - 4 = 26 - 4 2y = 22 y = 11 x = y + 4 = 11 + 4 = 15 Directions: Solve for the variables using either Elimination or Substitution Method.
 Q 1: Solve for the values of x and y for the following equations: 3x - y = 7 and 2x + 3y = 1.x = 2 and y = -1x = -2 and y = -1x = 2 and y = 1 Q 2: Solve for the values of x and y for the following equations: 2x + y = 6 and 2x + 2y = 24x = 18 and y = 6x = -6 and y = 18x = 6 and y = -18 Q 3: Solve for the values of x and y for the following equations: x + y = 10 and 3x + 2y = 24x = 6 and y = 4x = 2 and y = 3x = 4 and y = 6 Q 4: Solve for the values of x and y for the following equations: y = 3x + 8 and y = -xx = 4 and y = -2x = 2 and y = -2x = -2 and y = 2 Q 5: Solve for the values of x and y for the following equations: x + 3y = 7 and 2x + 5y = 12x = 1 and y = 2x = 2 and y = 2x = 1 and y = 1 Q 6: Solve for the values of x and y for the following equations: x + y = 6 and x - y = 4x = 5 and y = 1x = 4 and y = 2x = 1 and y = 5 Question 7: This question is available to subscribers only! Question 8: This question is available to subscribers only!