O Write The Answer Of The Following Questions. [Each Carr 36. Match Part-I With Part-II Properly: Part-I (i) X - Y = 5 (ii) 2x + 3y = 0 (iii) 3x - 2y = -2 (iv) X + 10 = 3y (v) 4x - 7y = 1 Part-I (a) (-7, 1) (b) (2, 1) (c) (-3, 2) (d) (7, 2) (e) (4, 7)
Introduction
Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will explore how to solve linear equations using various methods, including substitution and elimination. We will also provide a step-by-step guide on how to match the given linear equations with their corresponding solutions.
Understanding Linear Equations
A linear equation is an equation in which the highest power of the variable(s) is 1. It can be written in the form:
ax + by = c
where a, b, and c are constants, and x and y are variables.
Solving Linear Equations using Substitution
One method of solving linear equations is by substitution. This involves solving one equation for one variable and then substituting that expression into the other equation.
Example 1: Solving x - y = 5
Let's solve the equation x - y = 5 using substitution.
Step 1: Solve the equation for x.
x = y + 5
Step 2: Substitute the expression for x into the other equation.
2(x) + 3y = 0
2(y + 5) + 3y = 0
2y + 10 + 3y = 0
5y + 10 = 0
Step 3: Solve for y.
5y = -10
y = -2
Step 4: Substitute the value of y back into one of the original equations to solve for x.
x - y = 5
x - (-2) = 5
x + 2 = 5
x = 3
Therefore, the solution to the equation x - y = 5 is (3, -2).
Example 2: Solving 2x + 3y = 0
Let's solve the equation 2x + 3y = 0 using substitution.
Step 1: Solve the equation for x.
2x = -3y
x = (-3/2)y
Step 2: Substitute the expression for x into the other equation.
x - y = 5
(-3/2)y - y = 5
(-5/2)y = 5
y = -2
Step 3: Substitute the value of y back into one of the original equations to solve for x.
x - y = 5
x - (-2) = 5
x + 2 = 5
x = 3
Therefore, the solution to the equation 2x + 3y = 0 is (3, -2).
Example 3: Solving 3x - 2y = -2
Let's solve the equation 3x - 2y = -2 using substitution.
Step 1: Solve the equation for x.
3x = 2y - 2
x = (2/3)y - 2/3
Step 2: Substitute the expression for x into the other equation.
x + 10 = 3y
((2/3)y - 2/3) + 10 = 3y
(2/3)y - 2/3 + 10 = 3y
(2/3)y + 28/3 = 3y
(2/3)y - 3y = -28/3
(-5/3)y = -28/3
y = 28/5
Step 3: Substitute the value of y back into one of the original equations to solve for x.
x + 10 = 3y
x + 10 = 3(28/5)
x + 10 = 84/5
x = 84/5 - 10
x = 14/5
Therefore, the solution to the equation 3x - 2y = -2 is (14/5, 28/5).
Example 4: Solving x + 10 = 3y
Let's solve the equation x + 10 = 3y using substitution.
Step 1: Solve the equation for x.
x = 3y - 10
Step 2: Substitute the expression for x into the other equation.
2x + 3y = 0
2(3y - 10) + 3y = 0
6y - 20 + 3y = 0
9y - 20 = 0
Step 3: Solve for y.
9y = 20
y = 20/9
Step 4: Substitute the value of y back into one of the original equations to solve for x.
x + 10 = 3y
x + 10 = 3(20/9)
x + 10 = 60/3
x = 50/3
Therefore, the solution to the equation x + 10 = 3y is (50/3, 20/9).
Example 5: Solving 4x - 7y = 1
Let's solve the equation 4x - 7y = 1 using substitution.
Step 1: Solve the equation for x.
4x = 7y + 1
x = (7/4)y + 1/4
Step 2: Substitute the expression for x into the other equation.
x - y = 5
((7/4)y + 1/4) - y = 5
(7/4)y + 1/4 - y = 5
(-3/4)y + 1/4 = 5
(-3/4)y = 5 - 1/4
(-3/4)y = 19/4
y = -19/3
Step 3: Substitute the value of y back into one of the original equations to solve for x.
x - y = 5
x - (-19/3) = 5
x + 19/3 = 5
x = 5 - 19/3
x = 8/3
Therefore, the solution to the equation 4x - 7y = 1 is (8/3, -19/3).
Matching the Equations with their Solutions
Now that we have solved the linear equations, let's match them with their corresponding solutions.
| Equation | Solution |
|---|---|
| x - y = 5 | (3, -2) |
| 2x + 3y = 0 | (3, -2) |
| 3x - 2y = -2 | (14/5, 28/5) |
| x + 10 = 3y | (50/3, 20/9) |
| 4x - 7y = 1 | (8/3, -19/3) |
Therefore, the correct matches are:
- (i) x - y = 5 -> (a) (-7, 1)
- (ii) 2x + 3y = 0 -> (a) (-7, 1)
- (iii) 3x - 2y = -2 -> (e) (4, 7)
- (iv) x + 10 = 3y -> (d) (7, 2)
- (v) 4x - 7y = 1 -> (c) (-3, 2)
Note that the correct matches are (a) (-7, 1), (e) (4, 7), (d) (7, 2), and (c) (-3, 2).