Solving equations and inequalities
Mathematics, Grade 8
Solving equations and inequalities
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Solving equations and inequalities
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Study Guide Solving equations and inequalities Mathematics, Grade 8
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SOLVING EQUATIONS AND INEQUALITIES Algebraic equations are mathematical equations that contain a letter or variable which represents a number. • To solve an algebraic equation, inverse operations are used. • Multi-step equations involve many different mathematical operations that must be evaluated in order to solve the equation. • Equations can be solved using addition, subtraction, multiplication and division. Algebraic inequalities are mathematical equations that compare two quantities using greater than, >; g reater than or equal to, ≥; less than, <; a nd less than or equal to, ≤. • Multi-step inequalities are solved by using inverse operations similar to equations. • With inequalities, attention must be given when multiplying or dividing by a negative number. When this occurs, the inequality sign is reversed from the original inequality sign in order for the inequality to be correct. Equations and inequalities can contain more than one variable; a linear equation is such an example. Sometimes, an equation will need to be solved for a variable in terms of the other variable. In this case, the other variable is not solved for, but left alone and the equation is solved for the variable stated. Systems of equations are two linear equations can be solved for an ordered pair that will make both equations true. System of equations can be solved by using algebraic substitution, the addition method or the subtraction method. © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.
How to use solving equations and inequalities To evaluate multi-step equations and inequalities, inverse operations are used. • With multi-step equations and inequalities, it is very important to isolate the variable before evaluating. • Isolating the variable means to get the variable alone on one side of the equation. • The only rule that must be remembered when evaluating inequalities is that when a variable is multiplied or divided by a negative number the sign is reversed. Equations can contain more than one variable as seen in linear equations. Sometimes an equation with more than one variable will be solved for one variable in terms of the other variable. In this case, the equation is solved for the variable stated in terms the other variable. • For example, solve the linear equation, 6x + 2y = 8, for y in terms of x. Ex. 6x + 2y = 8 2y = -6x + 8 y = (-6/2)x + (8/2) → y = -3x + 4 This linear equation is solved for y in terms of x. This means that y is solved for and x is still in the equation. The equation should be in simplified form as shown. Another use of equations is called systems of equations. Systems of equations are two linear equations can be solved for an ordered pair that will make both equations true. © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.
• Solving for systems of equations can be done by algebraic substitution. Algebraic substitution means that one equation is substituted in the other equation. Consider the equations, y = 2x and 3x + y = 35, solving with algebraic substitution is as follows: Ex. y = 2x 3x + y = 35 y = 2x 3x + (2x) = 35 y = 2(7) 5x = 35 y = 14 x = 7 Once one variable is found, it is substituted back into one of the equations to find the other variable. Another method of solving systems of equations is by using the addition method or subtraction method. • These methods work when one variable term has the same coefficient in both equations. Then the equations are written vertically and either added or subtracted to make the variable with the same coefficients cancel out. This will leave only one variable in the equation to be solved. When the variable is solved, the answer is plugged into one of the original equations to get the value of the other variable. • One thing to remember with the subtraction method is to distribute the negative sign in the second equation in order to get the correct answer. © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.
Try This! 1. Solve for x, x/6 + 21 = 27 2. Solve for x, 9 - 5x = -31 3. Solve for x, 8x - 10 < 46 4. Solve for x, 7 - 3x ≤ 43 5. Solve for x, x/-5 - 9 > - 5 6. Solve for y in terms of x for the equation, 8x + 4y = 28 7. Solve for y in terms of x for the equation, 2x - 3y = -18 8. What is the solution set for the equations, y = 5x and y - 3x = 26 using algebraic substitution? 9. What is the solution set for the equations, 2x - 3y = 14 and 5x + 3y = 21 using the addition method? 10. What is the solution set for the equations, 6x + 8y = 72 and 3x + 8y = 30 using the substitution method? © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.
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