A system of linear equations is just a set of two or more linear equations. Show
In two variables ( x and y ) , the graph of a system of two equations is a pair of lines in the plane. There are three possibilities:
Zero solutions: y = − 2 x + 4 y = − 2 x − 3 One solution: y = 0.5 x + 2 y = − 2 x − 3 Infinitely many solutions: y = − 2 x − 4 y + 4 = − 2 x
There are a few different methods of solving systems of linear equations:
See the second graph above. The solution is where the two lines intersect, the point ( − 2 , 1 ) . Example 1: Solve the system { 3 x + 2 y = 16 7 x + y = 19 Solve the second equation for y . y = 19 − 7 x Substitute 19 − 7 x for y in the first equation and solve for x . 3 x + 2 ( 19 − 7 x ) = 16 3 x + 38 − 14 x = 16 − 11 x = − 22 x = 2 Substitute 2 for x in y = 19 − 7 x and solve for y . y = 19 − 7 ( 2 ) y = 5 The solution is ( 2 , 5 ) .Example 2: Solve the system { 4 x + 3 y = − 2 8 x − 2 y = 12 Multiply the first equation by − 2 and add the result to the second equation. − 8 x − 6 y = 4 8 x − 2 y = 12 _ − 8 y = 16 Solve for y . y = − 2 Substitute for y in either of the original equations and solve for x . 4 x + 3 ( − 2 ) = − 2 4 x − 6 = − 2 4 x = 4 x = 1 The solution is ( 1 , − 2 ) . Solving EquationsSolving equations involves finding the value of the unknown variables in the given equation. The condition that the two expressions are equal is satisfied by the value of the variable. Solving a linear equation in one variable results in a unique solution, solving a linear equation involving two variables gives two results. Solving a quadratic equation gives two roots. There are many methods and procedures followed in solving an equation. Let us discuss the techniques in solving an equation one by one, in detail.
What is the Meaning of Solving Equations?Solving equations is computing the value of the unknown variable still balancing the equation on both sides. An equation is a condition on a variable such that two expressions in the variable have equal value. The value of the variable for which the equation is satisfied is said to be the solution of the equation. An equation remains the same if the LHS and the RHS are interchanged. The variable for which the value is to be found is isolated and the solution is obtained. Solving an equation depends on what type of equation that we are dealing with. The equations can be linear equations, quadratic equations, rational equations, or radical equations. Steps in Solving an EquationThe aim of solving an equation is to find the value of the variable that satisfies the condition of the equation true. To isolate the variable, the following operations are performed still balancing the equation on both sides. By doing so LHS remains equal to RHS, and eventually, the balance remains undisturbed throughout.
After performing this systematic balancing method of solving an equation by a series of identical arithmetical operations on both sides of the equation, we separate the variable on one of the sides and the ultimate step is the solution of the equation. Solving Equations of One VariableA linear equation of one variable is of the form ax + b = 0, where a, b, c are real numbers. The following steps are followed while solving an equation that is linear.
Consider this example: 3(x + 4) = 24 + x We simplify the LHS using the distributive property. 3x + 12 = 24 + x Group the like terms together using the transposing method. This becomes 3x - x = 24-12 Simplify further ⇒ 2x = 12 Use the division property of equality, 2x/2 = 12/2 isolate the variable x. x = 6 is the solution of the equation. Use any one of the following techniques to simplify the linear equation and solve for the unknown variable. The trial and error method, balancing method and the transposing method are used to isolate the variable. Solving an Equation by Trial And Error MethodConsider 12x = 60. To find x, we intuitively try to find that 12 times what number is 60. We find that 5 is the required number. Solving equations by trial and error method is not always easy. Solving an Equation by Balancing MethodWe need to isolate the variable x for solving an equation. Let us use the separation of variables method or the balancing method to solve it. Consider an equation 2x + 3 = 17. We first eliminate 3 in the first step. To keep the balance while solving the equation, we subtract 3 from either side of the equation. Thus 2x + 3 - 3 = 17 - 3 We have 2x = 14 Now to isolate x, we divide by 2 on both sides. (Division property of equality) 2x/2 = 14/2 x = 7 Thus, we isolate the variable using the properties of equality while solving an equation in the balancing method. Solving an Equation by Transposing MethodWhile solving an equation, we change the sides of the numbers. This process is called transposing. While transposing a number, we change its sign or reverse the operation. Consider 5y + 2 = 22. We need to find y, so isolate it. Hence we transpose the number 2 to the other side. The equation becomes, 5y = 22-2 5y = 20 Now taking 5 to the other side, we reverse the operation of multiplication to division. y = 20/5 = 4 Solving an Equation That is QuadraticThere are equations that yield more than one solution. Quadratic polynomials are of degree two and the zeroes of a quadratic polynomial represent the quadratic equation. Consider (x+3) (x+2)= 0. This is quadratic in nature. We just equate each of the expressions in the LHS to 0. Either x+3 = 0 or x+2 =0. We arrive at x = -3 and x = -2. A quadratic equation is of the form ax2 + bx + c = 0. Solving an equation that is quadratic, results in two roots: α and β. Steps involved in solving a quadratic equation are:
By Completing The Squares MethodSolving an equation of quadratic type by completing the squares method is quite easy as we apply our knowledge of algebraic identity: (a+b)2
For more information about solving equations (quadratic) by completing the squares, click here. By Factorization MethodSolving an equation of quadratic type using the factorization method, follow the steps discussed here. Write the given equation in the standard form and by splitting the middle terms, factorize the equation. Rewrite the equation obtained as a product of two linear factors. Equate each linear factor to zero and solve for x. Consider 2x2 + 19x + 30 =0. This is of the standard form ax2 + bx + c = 0. Split the middle term in such a way that the product of the terms should equal the product of the coefficient of x2 and c and the sum of the terms should be b. Here the product of the terms should be 60 and the sum should be 19. Thus, split 19x as 4x and 15x (as the sum of 4 and 15 is 19 and their product is 60). 2x2 + 4x + 15x + 30 = 0 Take the common factor out of the first two terms and the common factors out of the last two terms. 2x(x+ 2) + 15(x + 2) = 0 Factoring (x+2) again, we get (x+ 2)(2x + 15) = 0 x = -2 and x = -15/2 Solving an equation that is quadratic involves such steps while splitting the middle terms on factorization. By Formula MethodSolving an equation of quadratic type using the formula x = [-b ± √[(b2 -4ac)]/2a helps us find the roots of the quadratic equation ax2 + bx + c = 0. Plugging in the values of a, b, and c in the formula, we arrive at the solution. Consider the example: 9x2 -12 x + 4 = 0 a= 9, b = -12 and c = 4 x = [-b ± √[(b2 -4ac)]/2a = [12 ± √[((-12)2 -4×9×4)] / (2 × 9) = [12 ± √(144 - 144)] / 18 = (12 ± 0)/18 x = 12/18 = 2/3 Solving an Equation That is RationalAn equation with at least one polynomial expression in its denominator is known as the rational equation. Solving an equation that is rational involves the following steps. Reduce the fractions to a common denominator and then solve the equation of the numerators. Consider x/(x-1) = 5/3 On cross-multiplication, we get 3x = 5(x-1) 3x = 5x - 5 3x - 5x = - 5 -2x = -5 x = 5/2 Solving an Equation That is RadicalAn equation in which the variable is under a radical is termed the radical equation. Solving an equation that is a radical involves a few steps. Express the given radical equation in terms of the index of the radical and balance the equation. Solve for the variable. Consider √(x+1) = 4 Now square both the sides to balance it. [ √(x+1)]2 = 42 (x+1) = 16 Thus x = 16-1 =15 Important Notes on Solving Equations:
☛ Related Articles:
FAQs on Solving EquationsWhat is Solving an Equation?Solving an equation is finding the value of the unknown variables in the given equation. The process of solving an equation depends on the type of the equation. What are The Steps in Solving Equations?Identify the type of equation: linear, quadratic, logarithmic, exponential, radical or rational.
What are The Golden Rule in Solving an Equation?The type of the equation is identified. If it is a linear equation, separating the variables method or transposing method is used. If it is a quadratic equation, completing the squares, splitting the middle terms using factorization is used or by formula method. How Do You use 3 Steps in Solving an Equation?The 3 steps in solving an equation are to
How Do You Solve Linear Equations?While solving an equation that is linear, we isolate the variable whose value is to be found. We either use transposing method or the balancing method. How Do You Solve Quadratic Equations?While solving an equation that is quadratic, we write the equation in the standard form ax2 + bx + c = 0, and then solve using the formula method or factorization method or completing the squares method. How Do You Solve Radical Equations?While solving an equation that is radical, we remove the radical sign, by raising both the sides of the equation to the index of the radical, isolate the variable and solve for x. How Do You Solve Rational Equations?While solving an equation that is rational, we simplify the expression on each side of the equation, cross multiply, combine the like terms and then isolate the variable to solve for x. What are the two methods to solve equation?There are three ways to solve systems of linear equations in two variables: graphing. substitution method. elimination method.
What are 2 step equations called?Two step equations are algebraic equations and are the equations that can be solved in exactly two steps and gives the final value of the variable in two steps. Generally, two step equations are of the form ax + b = c, where a, b, c are real numbers. A few examples of two step equations are: 2x + 3 = 7.
What are the two types of equations?There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variables. A conditional equation is only true for particular values of the variables. An equation is written as two expressions, connected by an equals sign ("=").
What are the 2 algebraic methods?The most-commonly used algebraic methods include the substitution method, the elimination method, and the graphing method.
|