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It is important to learn this process, so that you can apply it to more difficult equations! Solving Two-Step Equations To solve an equation, we must isolate the variable (get the variable on one side all by itself.) Every equation has 2 sides, separated by the = sign. left side 4 + x = 12 right side The solution to an equation is expressed as an equation with the variable ALONE on one side. The solution is : y = 2 example: y + 3 = 5 To islolate the variable, we use what we know about the IDENTITY properties. We know that x - 0 = x We know that We know that 1x = x. We know that x + 0 = x x 1 = x • Each side of an equation can be multiplied by the same non-zero number without changing the solution. • Each side of an equation can be divided by the same non-zero number without changing the solution. • The same number can be subtracted from both sides of the equation without changing the solution. • The same number can be added to both sides of the equation without changing the solution. We also use the PROPERTIES of EQUALITY Think of these properties as the Golden Rule: 3 = 3 3 + 2 = 3 + 2 What you do to one side of the equation, you MUST do to the other. 4 = 4 4(6) = 4(6) 7 = 7 7 - 5 = 7 - 5 16 = 16 16÷8 = 16÷8 We also use INVERSE OPERATIONS. The inverse of addition is subtraction. The inverse of subtraction is addition. The inverse of multiplication is division. The inverse of division is multiplication. Example 1 The inverse of addition is subtraction, so we need to subtract 9 from both sides of the equation. We need to UNDO the 9, which is added to p. Inverse and Equality We want to isolate the variable p. Identity p + 9 = 12 p + 0 = 3 p - 9 = - 9 3 You must show the steps of solving your equations in this manner! Example 2 We need to UNDO the 3, which is subtracted from s. We want to isolate the variable s. Inverse and Equality The inverse of subtraction is addition, so we need to add 3 to both sides of the equation. Identity s s - 0 = 20 s - 3 = 17 + 3 = + 3 20 You must show the steps of solving your equations! Example 3 We need to UNDO the 5, which is multiplied by x. We want to isolate the variable x. Inverse and Equality The inverse of multiplication is division, so we need to divide both sides of the equation by 5. Identity ÷ 1x = 7 5x = 35 5 x = ÷ 5 7 You must show the steps of solving your equations! We use this same process when more than one number is on the same side of the equation as the variable. However, it is important to remember to use inverse operations to
undo ADDITION or SUBTRACTION BEFORE undoing MULTIPLICATION or DIVISION.
Use inverse operations to undo the constant before undoing the coefficient. Coefficient 4 x + 9 = 21 Constant Example 1 4 4x + 0 = 12 x 4 4 + 4x = 12 - 9 1x 9 x = = = - 9 21 3 3 subtract 9 from both sides divide both sides by 4 solution Example 2 2p - 3 = 27 2p - 0 = 30 2 2p = 30 1p = 15 + 3 p = 15 +3 2 add 3 to both sides divide both sides by 2 solution Your turn! 5y - 15 = 20 5y - 0 = + Fill in the missing numbers. 5y = 35 1y y = = + 35 Fill in the missing operations and numbers. 9c + 3 = 39 9c + 0 = 9c = 1c 3 c = = 3 What is the first step in solving this equation? 6a - 3 = 21 add 6 add 3 subtract 6 subtract 3 Identify the next step in solving this equation. 4b + 8 = 56 4b + 0 = 48 4b = 48 - 8 add 4 to both sides multiply both sides by 4 divide both sides by 4 subtract 4 from both sides -8 Remember: FIRST, undo any addition or subtraction. What you do to one side of the equation, you must do to the other. SHOW your work on both sides! THEN, undo any multiplication. Use inverse operations to isolate the variable. |