# Algebra - Simultaneous Equations

Solve Simultaneous Equations

Simultaneous equations are a set of two or more equations with multiple variables. (This is often referred to as a system of equations.)

Solve the following two equations for x and y.

Use the substitution, elimination (addition/subtraction), or graphical method

y = 1/3x - 2, y = -1/3x +1

### Comments for Algebra - Simultaneous Equations

 Feb 11, 2013 Solve Simultaneous Equations by: Staff Answer Part I Use the substitution, elimination (addition/subtraction), or graphical method y = 1/3x - 2, y = -1/3x +1 Solving for x and y using substitution The goal of the substitution method is to eliminate one of the variables using substitution. y = ⅓ x - 2 y = -⅓ x +1 Since the 1st equation shows that y = ⅓ x - 2, substitute ⅓ x - 2 for y in the 2nd equation. ⅓ x - 2 = -⅓ x +1 The substitution transforms the 2nd equation into an equation which contains only one variable (the variable x). Now solve for x. ⅓ x - 2 = -⅓ x +1 Add ⅓ x to each side of the equation. ⅓ x - 2 + ⅓ x = -⅓ x + 1 + ⅓ x combine like terms ⅓ x + ⅓ x - 2 = -⅓ x + ⅓ x + 1 (⅓ x + ⅓ x) - 2 = (-⅓ x + ⅓ x) + 1 ⅔ x - 2 = 0 + 1 ⅔ x - 2 = 1 Multiply each side of the equation by 3 3 * (⅔ x - 2) = 3 * 1 3 * (⅔ x) + 3 * ( - 2) = 3 * 1 3 * (⅔ x) + 3 * ( - 2) = 3 * 1 2 * (3 / 3) * x + 3 * ( - 2) = 3 * 1 2 * (1) * x + 3 * ( - 2) = 3 * 1 2 x + 3 * ( - 2) = 3 * 1 2 x - 6 = 3 * 1 2 x - 6 = 3 Add 6 to each side of the equation 2 x - 6 + 6 = 3 + 6 2 x + 0 = 3 + 6 2 x = 3 + 6 2 x = 9 Divide each side of the equation by 2 2 x / 2 = 9 / 2 x * (2 / 2) = 9 / 2 x * (1) = 9 / 2 x = 9 / 2 x = 4.5 Now that you know that the value of x = 4.5, substitute 4.5 for x in either of the original two equations. It does not matter which equation you choose. y = ⅓ x - 2 y = ⅓ * 4.5 - 2 y = 1.5 - 2 y = -0.5 the final answer is: x = 4.5 y = - 0.5 -------------------------------------------------------

 Feb 11, 2013 Solve Simultaneous Equations by: Staff ------------------------------------------------------- Part II Solving for x and y using the elimination (addition/subtraction) method The reason for using the elimination method is that one of the variables can be eliminated by addition or subtraction. y = ⅓ x - 2, y = -⅓ x +1 Line up the two equations up vertically (one under the other). y = ⅓ x - 2 y = -⅓ x +1 The 1st equation contains the positive term ⅓ x. The second equation contains the term -⅓ x. When these two equations are added, these terms will cancel one another. This completely eliminates the x variable and allows you with only one equation with one variable (the variable y) . y = ⅓ x - 2 +(y = -⅓ x +1) ----------------------- (y + y) = (⅓ x - ⅓ x) + (- 2 + 1) 2y = 0 - 1 2y = - 1 The next step is to divide each side of the equation by 2 2y / 2 = - 1 / 2 y * (2 / 2) = - ½ y * (1) = - ½ y = - ½ Now that you know that the value of y = - ½, substitute - ½ for y in either of the original two equations. It does not matter which equation you choose. y = ⅓ x - 2 - ½ = ⅓ x - 2 You now have an equation containing only the variable x. Solve for x - ½ = ⅓ x - 2 multiply each side of the equation by 6 6 * (- ½) = 6 * (⅓ x - 2) - (6/2) = (6/3) x - (6 * 2) - 3 = 2 x - 12 2 x - 12 = - 3 Add 12 to each side of the equation 2 x - 12 + 12 = - 3 + 12 2 x + 0 = - 3 + 12 2 x = - 3 + 12 2 x = 9 Divide each side of the equation by 2 2 x / 2 = 9 / 2 x * (2 / 2) = 9 / 2 x * (1) = 9 / 2 x = 9 / 2 x = 4.5 the final answer is: x = 4.5 y = - 0.5 -------------------------------------------------------

 Feb 11, 2013 Solve Simultaneous Equations by: Staff ------------------------------------------------------- Part III Solving for x and y using the graphical method Plot both of the equations on the same set of x-y coordinates.: y = ⅓ x - 2 y = -⅓ x +1 The solution values of x and y is that point where both graphs intersect. Thanks for writing. Staff www.solving-math-problems.com