UNIQUE FRIENDS SCHOOLSLet's dive into solving simultaneous linear equations using the Elimination and Substitution methods, tailored for JSS 3 students following a hybrid curriculum.
Solve for x and y: 2x + 3y = 7 x - 2y = -3
Step 1: Multiply the two equations by necessary multiples such that the coefficients of y's in both equations are the same:
Multiply the first equation by 2 and the second equation by 3 to align the coefficients of y.
(2x + 3y = 7) * 2 => 4x + 6y = 14
(x - 2y = -3) * 3 => 3x - 6y = -9
Step 2: Add both equations to eliminate y.
(4x + 6y) + (3x - 6y) = 14 + (-9) 7x = 5
Step 3: Solve for x.
x = 5 / 7
Step 4: Substitute x back into one of the original equations to solve for y. Using the first equation:
2(5/7) + 3y = 7
Step 5: Solve for y.
10/7 + 3y = 7 3y = 7 - 10/7 3y = (49 - 10) / 7 3y = 39 / 7 y = 13 / 7
Solve for x and y: x + 2y = 4 3x - 2y = 6
Step 1: Solve one of the equations for one variable. Let's solve the first equation for x:
x = 4 - 2y
Step 2: Substitute the expression for x from Step 1 into the other equation:
3(4 - 2y) - 2y = 6
Step 3: Solve for y.
12 - 6y - 2y = 6 12 - 8y = 6 -8y = 6 - 12 -8y = -6 y = 6 / 8 y = 3 / 4
Step 4: Substitute y back into the equation x = 4 - 2y to find x:
x = 4 - 2(3/4) x = 4 - 3/2 x = (8 - 3) / 2 x = 5 / 2
Solve for x and y: 4x + 5y = 23 2x - 3y = 5
Step 1: Multiply the two equations by necessary multiples such that the coefficients of y's in both equations are the same:
To eliminate y, we need the coefficients in front of y to be the same but with opposite signs. Multiply the first equation by 3 and the second by 5.
(4x + 5y = 23) * 3 => 12x + 15y = 69
(2x - 3y = 5) * 5 => 10x - 15y = 25
Step 2: Add both equations to eliminate y.
(12x + 15y) + (10x - 15y) = 69 + 25 22x = 94
Step 3: Solve for x.
x = 94 / 22 x = 47 / 11
Step 4: Substitute x back into one of the original equations to solve for y. Using the first equation:
4(47/11) + 5y = 23
Step 5: Solve for y.
188/11 + 5y = 23 5y = 23 - 188/11 5y = (253 - 188) / 11 5y = 65 / 11 y = 13 / 11
For visual learners, consider a graph where these lines intersect. Each equation represents a line on the coordinate plane. The point of intersection represents the solution (x, y).
These textbooks offer comprehensive coverage of simultaneous linear equations, including both the Elimination and Substitution methods, with numerous examples and exercises for practice.
Solving simultaneous linear equations is a fundamental skill in mathematics, crucial for more advanced topics. By mastering the Elimination and Substitution methods, students can efficiently solve systems of equations, which is essential in various mathematical and real-world applications. Regular practice with a variety of examples will solidify understanding and build confidence.