Each arithmetical expression can be composed as a single term or a progression of terms separated by plus or minus signs.
For example:
A statement that equates two algebraic expressions is called an equation.
There are three types of equations. They are:
1. Linear equation in one variable.
2. Linear equation in two variables.
3. Quadratic equation in one variable.
1.Solving equations with one variable:-
To solve a linear equation in one variable intends to discover the estimation of the variable that makes the equation true. Two equations that have the same solution are said to be equivalent.
For example,
\(x\) + 2 = 4 and 4\(x\) + 1 = 2 are equivalent equations
Here, both are true when \(x\) = 1 and are false otherwise.
Two basic rules are important for solving linear equations:-
Example 1:
5\(x\) – 2 = 13
Solution:
Example 2:
Solve the equation: 4\(x\) = 8
Solution:
Example 3:
Solve: 3x=12
Solution:
2. Solving equations with two variables:-
To calculate linear equations in two variables, it is important to have two equations that are not proportionate. To explain such a “system” of concurrent equations, there are two basic methods.
3. Solving quadratic equations with one variable:-
A quadratic equation is any equation that can be expressed in the form of a\(x^2\) + b\(x\) + c = 0.
Where a, b,and c are real numbers (a ≠ 0).
Such an equation can be solved by the formula
\(x\) = \( \frac{-b ± \sqrt{b^2 – 4ac}}{2a}\)
Example:
2. \((x^a)(x^b)\) = \(x^{a + b}\)
3. \((x^a)(y^a)\) = \((xy)^a\)
4. \(\frac{x^a}{x^b}\) = \(x^{a – b}\) = \(\frac{1}{x^{a – b}}\) (\(x\) ≠ 0)
5. \(\frac{x^a}{y^a}\) = \((\frac{x}{y})^a\) (\(y\) ≠ 0)
6. \((x^a)^b\) = \(x^{ab}\)
7. If \(x\) ≠ 0, then \(x^0\) = 1
2. Solve 3\(z^2\) -5\(z\) + 2 = 0
Solution:
3. A gathering of students are at visit. The aggregate toll is Rs. 120 and this is to be shared similarly among the students. In the event that two more students join the visit, each will pay Rs. 2 less. find the original number of student in the gathering.
Solution:
4. Quantitative comparison of
\(x\) + \(y\) = 5
\(y\) – \(x\) = 3
Quantity A | Quantity B |
\(x\) | \(y\) |
A. The quantity on the left is greater.
B. The quantity on the right is greater.
C. Both are equal.
D. The relationship cannot be determined without further information.
Solution:
5. Quantitative comparison of
\(x^2\) – 12\(x\) + 35 = 0
Quantity A | Quantity B |
\(x\) | 12 |
A. The quantity on the left is greater.
B. The quantity on the right is greater.
C. Both are equal.
D. The relationship cannot be determined without further information.
Solution: