Math Equations are those where two algebraic expressions involving a one or more variables are equated. If the equation involves only one variable, it is called a linear equation in one variable. We need to have only one equation to solve the equation for the variable. IF the equation involves two or more variables, we need to have as many linear equations ( also called simultaneous linear equations ) as there are variables. The solutions to these equations give the point of intersection of the lines or the planes as the case may be. We can also find the points of intersection of the curves by solving the equations of the curves. In this section let us see the method of solving some of the math equations.

=> 3x + 5 - 5 = 29 - 5 [ adding both sides the additive inverse of 5 which is -5 ]

=> 3x = 24 [ multiplying both sides by the reciprocal of 3 which is 1/3 ]

=> $\frac{1}{3}$ ( 3x ) = $\frac{1}{3}$ ( 24 )

=> x = 8

Therefore the solution to the above equation is x = { 8 }

=> 8x - 34 = 3x - 9 [ multiplying by the integer outside the parenthesis ]

=> 8x - 3x - 34 = 3x - 9 - 3x [ subtracting 3x from both sides of the equation ]

=> 5x - 34 = - 9

=> 5x - 34 + 34 = -9 + 34 [ adding 34 on both sides of the equation ]

=> 5x = 25

=> $\frac{5x}{5}$ = $\frac{25}{5}$ [ dividing by 5 on both sides of the equation ]

=> x = 5

The linear equations of two variables can be solved if we have at least two equations containing the two variables.The pair equations containing the variables is also called as simultaneous equations of two variables. Because the pair of equations have a common solution which satisfy both the equations simultaneously.

Example : Check if ( 2, 3 ) is the solution of the equations, x + y = 5 and 2x - y = 1**Solution:** Let us plug in the values x = 2 and y = 3 in the two equations,

=> 2 + 3 = 5

=> 5 = 5

=> the point ( 2, 3 ) is the solution of the equation x + y = 1.

Let us plug in x = 2 and y = 3 in the equation, 2x - y = 1

=> 2 ( 2) - 3 = 1

=> 4 - 3 = 1

=> 1 = 1

=> ( 2, 3) is the solution of the equation 2x - y = 1

Therefore

We observe from the above graph that the two lines intersect at the point ( 2, 3 ).

Solving second degree equations: The graph of second degree equations is a curve. By solving the second degree equations we get the point of intersection of the two curves.

Example : Solution to the second degree equations x

Solution : We have the equations, x

and 16 x = 3 y

Multiplying Equation (1) by 3, we get,

3 x

= > 3 x

=> 3 x

=> 3 x

=> x ( 3x + 25 ) - 3 ( 3x + 25 ) = 0

=> ( x - 3 ) ( 3 x + 25 ) = 0

=> x = 3 or x = - 25 /3

Value of x cannot be negative, since the curve opens towards right.

Therefore x = 3. substituting x = 3 in 16 x = 3 y

16 ( 3 ) = 3 y

=> 3 y

=> y

=> y = $\pm$ 4

When we draw the graph using graphing calculator we get the graphs as shown below.

We observe from the above graph that the two curves intersect at the points ( 3, 4 ) and ( 3, -4 )

2. Solve 2 ( x + 4 ) = 5 ( 3 - x ), by 5 steps method.

3. Solve the simultaneous linear equations, y = 3x - 5 and 2x + 3y = 18 by substitution method.

4. Solve the second degree equations, $\left ( \frac{x^{2}}{9} \right )$ + $\left ( \frac{y^{2}}{16} \right )$ = 1

and $\left ( \frac{y^{2}}{9} \right )$ + $\left ( \frac{x^{2}}{16} \right)$ = 1

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