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Algebra 1 Problems

Algebra is the branch of Mathematics evolved with new concepts, when arithmetic is extended from Numbers to variables. The curriculum for Algebra I consists of fundamental topics in Algebra.
The practice of Algebra I problems is necessary to acquire strong analytical skills, a pre requisite for the study of higher level Math.
In Algebra I, the following topics are generally covered:

  1. Classification of real number system and the arithmetic operations on them.
  2. Algebraic expressions and statements.
  3. Linear Equations in one and two variables.
  4. Linear and absolute value inequalities.
  5. Introduction to Polynomials and Quadratic Equations.
  6. Rational and Radical Expressions.

Let us look into few example of commonly found Algebra I problems.

Parallel and Perpendicular Lines:
Finding the equations of parallel and perpendicular lines to a given line is a common problem given for practice. Let us solve one problem on this.

Example:

The equation of line l1 is 2x - y = 5. Find the equation of a line passing through (2, -3) and

  1. Parallel to l1.
  2. Perpendicular to l1.
Solution:

Writing the equation in slope intercept form as y = 2x - 5, the slope of l1 is 2.

The slopes of two parallel lines are equal. Hence, the slope of a line parallel to l1 is also 2.
Using the slope point formula, the equation of the parallel line passing through (2, -3) is
y - (- 3) = 2(x - 2)
y + 3 = 2x - 4
=> 2x - y = 7.

The product of slopes of two perpendicular lines = -1.
Hence, the slope of a line perpendicular to l1 will be the negative reciprocal of the slope of l1.
Thus, the slope of the perpendicular line = -$\frac{1}{2}$

Hence, the equation of the perpendicular line through (2, -3) is

y + 3 = -$\frac{1}{2}$ (x - 2)
   
On simplifying, we get
x + 2y = -4.

System of Linear Equations:
Let us solve an application problem on Linear Systems.

Example:

The total weight of a tin packed full with cookies is 2 pounds. After $\frac{3}{4}$ of cookies are eaten, the tin with remaining cookies weight 0.8 pounds. Find the weight of the empty tin.

Solution:

Let the weight of the empty tin = x pounds and the weight of the cookies, when the tin is full = y pounds.
The two equations formed are,
x + y = 2                       ---------(1)                     Weight of fully packed tin.
x + 0.75y = 0.8              ---------(2)                     Weight after eating $\frac{3}{4}$ of cookies.

The above system of equations can be solved by elimination. Subtracting equation (2) from (1)
0.75 y = 1.2

y = $\frac{1.2}{0.75}$ = 1.6

Weight of empty tin, x = 2 - y = 2 - 1.6 = 0.4
Hence, the weight of empty tin is 0.4 pounds. 

Solving Quadratic Equations:
Let us work out an example problem on solving quadratic equations using the method of completing the square.

Let us consider the equation x2 - 4x - 12 = 0
x2 - 4x  = 12     (Constant is isolated)
x2 - 4x + (-2)2 = 12 + (-2)2      (Square of half of the coefficient of x is added to complete the square)
(x - 2)2 =  16
$\sqrt{(x-2)^{2}}$ = $\pm$ $\sqrt{16}$      (Square root is taken over the equation)
x - 2 = $\pm$ 4
x - 2 = 4  or   x - 2 = -4
x = {-2, 6}

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