Graph: The graph of an equation in x and y is locus of all points (x, y) satisfied by the graph of linear equation will always equation be straight line since a straight line is determined by two points for a line.
Graph of a linear equation in one variable: The graph of a linear equation in only unknown quantity is either (i) the x-axis (ii) the y-axis (iii) a line parallel to one of the axes. Thus, (a) the graph of y = 0 is the x-axis and the graph x = 0 is the y-axis. (b) the graph of y = a is a line parallel to the x-axis (c) the graph of x = b ·is a line parallel to the y-axis.
Equation: An equation is a statement of equality which involves one or more unknown quantities called the variable.
Linear equation: An equation involving only linear polynomials is called a linear equation.
Equation solution of equation: A value of the variable, which when substituted for the variable in the equation, makes the two sides (of the equation) equal, is called the solution of the equation.
Rules for solving a linear equation:
(i) Same number can be added to both sides of the equation.
(ii) Same number can be subtracted from both sides of the equation.
(iii) Both sides of the equation can be multiplied by the same number.
(iv) Both sides of the equation can be divided by the same non-zero number.
Note: Any term of an equation may be taken to the other side with the sign changed. The process is called transposition.
If ( ax+b / cx+d ) = p/q
then q(ax + b) = p(cx + d).
This process is called cross multiplication.
Applications of linear equations:
Basically, we follow three steps to solve problems based on linear equations :
1. Denote the unknown quantity by x.
2. From the information given, formulate a linear equation for x.
3. Solve the linear equation to find x.
Note: If there are more than one unknown quantities than in step 1, call one of these as x and write others in terms of x.
RULES FOR SOLVING LINEAR EQUATIONS IN TWO VARIABLES
Linear equation in two variables:
An equation of the form ax + by + c = 0, where a, b, c Î R, a ¹ 0, b ¹ 0, and x, y are two variables is called linear equation in two variables.
Solution: Any pair of values of x and y which satisfies the equation ax + by + c = 0 where a ¹ 0, b ¹ 0, a, b, c Î R is called its solution.
1. From either of the given equations, we express one of the two unknowns in terms of the other.
2. We substitute the value of unknown thus expressed in the other equation. By solving this equation, we get the value of one unknown.
3. We substitute the value of this unknown in either of the equations. By solving that, we get the value of the other unknown.
(a) We multiply both the equations by such numbers so as to make the coefficient of one of the two unknowns numerically the same.
(b) Then we add or subtract so as to get an equation containing only the other unknown. By solving these equations, we get the value of the one unknown.
(c) We substitute the value of this unknown in either of the equations. By solving that, we get the value of the other unknown.
Steps for drawing the graph of a linear equation:
(1) Rewrite the equation with y as the subject.
(2) Draw the value table for x and y by taking any three convenient values for x, find the corresponding value of y.
(3) Draw the axes on the squared paper and mark the scale.
(4) Plot three ordered pairs from the table.
(5) Join the three plots by using a straight edge and produce on either to run completely across the paper.
(6) While tabulating the values for x and y, values should be taken to represent positive and negative values. It should be ensured that the points that are plotted are not too close.
(1) Simultaneous linear equations in two variables: When two or more equations are satisfied by the same set of value of two variables.' then they are known as simultaneous equations. The pair of values as x and y satisfying each one of the given equations is called a solution of the system.
(2) Consistent and inconsistent systems of simultaneous linear equations: A system consisting of two simultaneously linear equations is said to be consistent, if it has at least one solution. Otherwise, if there does not exist any solution of the system, it is said to be inconsistent.
(3) Rules for solving simultaneous linear equations: There are three methods for solving simultaneous equations
- Method of substitution,
- Method of elimination,
- Method of cross-multiplication.
Note: If in ax + by + c = 0, we have a = 0, b ¹ 0 or b = 0, a ¹ 0 then equation reduces to by + c = 0 or ax + c = 0 respectively. In either case we get an equation in one variable. That is why both a and b are assumed to be non-zero.
(4) Graphical method of solution of simultaneous equations: Let the lines 1 and m represent the graphs of two equations respectively. The following three possibilities arise:
(1) Exactly one solution, if l and m (i.e., graphs) are intersecting lines;
(2) Infinitely many solutions, if l and m (i.e., graphs) are coincident; and
(3) No solution, if l and m (i.e., graphs) are parallel lines.
To solve a problem, we follow three steps: Make assumptions, using two variables, say, x and y.
From the information given, formulate two equations in terms of x and y.
Solve the equations simultaneously and verify the results.
Properties of equality:
(1) We can add the same quantities to both sides of an equality without changing it.
(2) We can subtract the same quantity from both sides of an equality without changing it.
(3) We can multiply both sides of an equality by the same number without changing it.
(4) We can divide both sides of an equality by the same non-zero number without changing it.
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