Solving an Equation

Pronunciation: /sɒlv iŋ ən ɪˈkweɪ ʒən/ ?

Most math problems you will find in algebra involve solving equations. The idea is to find a solution to the equation. A solution is a set of one or more values that, when substituted for the variable(s), make the equation true.

Solving equations is one of the most important math skills to learn. There are several classes of equations you will encounter in your studies:

  • Single variable equations of degree one such as x+3 = 2x-4. The goal in solving these equations is to get the variable by itself on one side of the equals sign. What is left on the other side of the equation is the value of the variable.
  • Single variable equations of degree two such as 2x2-3x+4 = 0. The quadratic formula is used for these quadratic equations.
  • Two variable equations of degree one such as y = x-4. These equations are linear equations. They can be graphed as a line. Linear equations with two variables have infinite solutions.
  • Two equations of two variables, each being degree one such as y = 3x + 7, y=-x-4. This is called a linear system. Linear systems have either no solution, one solution, or infinite solutions.
There are other types of equations to solve you may encounter in more advanced classes. They will not be discussed here.

Article Index

What Does it Mean to Solve an Equation?
Properties of Numbers Used to Solve Equations
empty spaceUsing Addition and Subtraction to Solve an Equation
empty spaceUsing Multiplication to Solve an Equation
empty spaceUsing Division to Solve an Equation
empty spaceUsing the Distributive Property to Solve an Equation
Solving an Equation Graphically
Solving a Quadratic Equation
Solving a Multiple Variable Equation

What Does it Mean to Solve an Equation?

To solve an equation is to find out what the equation tells one about the possible values of one or more variables. Take the equation x+3 = 2x-1. What does this equation tell one about the value of x? To find out, solve the equation.

Solving a one variable linear equation involves getting the variable on one side of the equals sign by itself. To do this one uses the properties of numbers.

Properties of Numbers Used to Solve an Equation

PropertyExampleDescription
Property of adding zeroGiven a real number x; x+0 = xZero added to any number equals the original number.
Addition property of equalityGiven three real numbers a, b, and c; if a=b then a+c = b+cIf two numbers are equal, after adding the same value to each number, the sum remain equal.
Subtraction property of equalityGiven three real numbers a, b, and c; if a=b then a-c = b-cIf two numbers are equal, after subtracting the same value from each number, the difference remain equal.
Property of multiplying by 1Given a real numbers x; 1·x = xAny number multiplied by 1 equals itself.
Distributive property of multiplication over addition and subtractionGive 3 real numbers a, b, and c; a(b+c) = ab+acA number multiplied by a sum is equal to the sum of the number multiplied by each term.
Transitive property of equalityGiven three real numbers a, b, and c; if a=b and b=c then a=cIf two numbers are equal to the same other number, they are equal to each other.
Table 1: Properties of numbers used to solve an equation

Using Addition and Subtraction to Solve an Equation

Start with the equation x+3 = 5. Since the goal is to get x by itself, one needs to do something about the 3 that is added to the x. To undo addition, use subtraction:

StepEquationDescription
1x+3 = 5Original equation
2(x+3)-3 = 5-3Subtraction property of equality.
3(x+3)+(-3) = 5-3Definition of subtraction.
4x+(3+(-3)) = 5-3Associative property of addition.
5x+0 = 2Simplify both sides of the equation.
6x = 2Property of addition by zero.
7(2)+3 = 5
5 = 5
Check work by substituting the solution back into the equation.
Table 2: Solving x+3 = 5.

Now start with the equation x-2 = 1. Since addition undoes subtraction, use the addition property of equality.

StepEquationDescription
1x-2 = 1Original equation
2(x-2)+2 = 1+2Addition property of equality.
3(x+(-2))+2 = 1+2Definition of subtraction.
4x+((-2)+2) = 1+2Associative property of addition.
5x+0 = 3Simplify both sides of the equation.
6x = 3Property of addition by zero.
7(3)-2 = 1
1 = 1
Check work by substituting the solution back into the equation.
Table 3: Solving x-2 = 1.

Using Multiplication to Solve an Equation

Start with the equation x÷3 = 6. Again, one wants to get the x by itself on one side of the equation. To undo multiplication, use division:

StepEquationDescription
1x÷3 = 6Original equation
2(x÷3)·3 = 6·3Multiplication property of equality.
3x·(3÷3) = 6·3Associative property of multiplication.
4x·1 = 18Simplify both sides of the equation.
5x = 18Property of multiplication by 1.
618÷3 = 6
6 = 6
Check work by substituting the solution back into the equation.
Table 4: Solving x÷3 = 6.

Using Division to Solve an Equation

When using division to solve an equation, one must be sure to not divide by zero. Since division by zero is undefined, division by zero will give undesirable and incorrect results. Start with the equation 3x = 6.

StepEquationDescription
13x = 6Original equation
2(3x)÷3 = 6÷3Division property of equality.
3(x·3)÷3 = 6÷3Commutative property of multiplication.
4x·(3÷3) = 6÷3Associative property of multiplication.
5x·1 = 2Simplify both sides of the equation.
6x = 2Property of multiplication by 1.
73·2 = 6
6 = 6
Check work by substituting the solution back into the equation.
Table 5: Solving 3x = 6.

Using the Distributive Property to Solve an Equation

The distributive property of multiplication over addition and subtraction is used often to solve an equation. The example in table 6 uses the distributive property to remove parenthesis.

StepEquationDescription
13(x-4) = 3Original equation
23·x-3·4 = 3Distributive property of multiplication over addition and subtraction.
33x-12 = 3Simplify.
4(3x-12)+12 = 3+12Addition property of equality.
5(3x+-12)+12 = 3+12Definition of subtraction.
63x+(-12+12) = 3+12Associative property of addition.
73x+0 = 15Simplify.
83x = 15Property of addition by zero.
9x·3 = 15Commutative property of multiplication
10x·3÷3 = 15÷3Division property of equality.
11x·1 = 5Simplify both sides of the equation.
12x = 5Property of multiplication by 1.
133(5-4) = 3
3·1 = 3
3 = 3
Check work by substituting the solution back into the original equation.
Table 6: Solving 3(x-4) = 3.

The example in table 7 uses the distributive property to combine two terms containing the same variable.

StepEquationDescription
13x = 2x + 3Original equation
23x - 2x = 2x - 2x + 3Addition property of equality.
33x - 2x = 0 + 3Simplify.
43x - 2x = 3Property of addition by zero.
5x(3 - 2) = 3Distributive property of multiplication over addition and subtraction.
6x·1 = 3Simplify.
7x = 3Property of multiplying by 1.
83·3 = 2·3 + 3
9 = 6 + 3
9 = 9
Check work by substituting solution back into the equation.
Table 7: Solving 3x = 2x + 3.

Solving an Equation Graphically

This section describes how to use a TI-83 or TI-84 graphing calculator to find the middle root of x3-0.5x2-2x+1. Most graphing calculators have a way to solve equations graphically. Check the user manual for your calculator.

StepScreenDescription
1TI-83/84 calculator blank screen.Click the <CLEAR> button until you have a blank screen.
2TI-83/84 calculator screen for inputing equations to graph. There are no equations showing.Click the <Y=> button on the top row to view the graphing equation screen. If there are any equations showing, use the <CLEAR> key and the up and down arrows to erase all equations. Then click the up arrow key until it is in the top position.
3TI-83/84 calculator screen for inputing equations to graph. The equation x^3-0.5x^2-2x+1 is in the Y1= row.Enter the equation x3-0.5x2-2x+1 using the following keystrokes: <X,T,θ,n>, <^>, <3>, <->, <0>, <.>, <5>, <X,T,θ,n>, <x2>, <->, <2>, <x>, <+>, <1>.
4TI-83/84 calculator screen for entering graphing window parameters.Click on the <WINDOW> button to load the graphing window parameter screen. Use the number keys and the up and down arrow buttons to enter the window parameters as show in the screen image. When entering negative numbers, make sure to use the <(-)> key on the bottom row of the calculator. The key labeled <-> means subtract, not negate.
5TI-83/84 calculator screen plot screen showing a graph of the equation y=x^3-0.5x^2-2x+1.Click the <GRAPH> button on the top row of the calculator. The graph of x3-0.5x2-2x+1 will appear in the window you specified.
6TI-83/84 calculator zoom menu.Click the <ZOOM> button to see the zoom menu.
7TI-83/84 calculator zoom menu with option 2-Zoom In selected.Click the down arrow button once to select menu option 2-Zoom In.
8TI-83/84 calculator with equation x^3-0.5x^2-2x+1 graphed and the zoom option activated.Click the <ENTER> key on the bottom right corner of the calculator to return to the graph with the zoom option activated.
9TI-83/84 calculator with equation x^3-0.5x^2-2x+1 graphed and the zoom option activated. The zoom coordinates are x=.55319149 and y=0.Click the right arrow button until the cross hairs are near the middle intercept. The numbers at the bottom of the screen change to show you the coordinates of the cross hair.
10TI-83/84 calculator with equation x^3-0.5x^2-2x+1 graphed and the zoom option activated. The zoom coordinates are x=.55319149 and y=0. The graph has zoomed in on the cross hairs.Click the <ENTER> button to zoom in on the zoom coordinates.
11TI-83/84 calculator with equation x^3-0.5x^2-2x+1 graphed and the zoom option activated. The zoom coordinates are x=.5 and y=0.Click on the left arrow button until the cross hairs are over the intercept. The zoom coordinates should be x=.5 and y=0. The correct intercept is x=0.5. You can click on the enter key again to make sure you are as close to the intercept as possible.
Table 8: Using the TI-83/84 calculator to find a root of x3-0.5x2-2x+1.

Solving a Quadratic Equation

Quadratic equations in the form ax2 + bx + c = 0 are solved using the quadratic formula:
x=(-b+-square root(b^2-4ac))/(2a) a!=0

The a, b and c in the quadratic formula are the a, b and c in the quadratic equation. Because of the ± in the equation, a quadratic equation can have two solutions.

If the discriminant b2 - 4ac is greater than zero, the quadratic equation has two real solutions. A real solution is a solution that is a real number. If the discriminant b2 - 4ac is 0, the quadratic equation has one real solution. If the discriminant b2 - 4ac is less than zero, the quadratic equation has two complex solutions. A complex solution is a solution that is a complex number.

Solving Using the Quadratic Equation
StepEquationDescription
empty space2x^2-4x-2=0Equation to solve
1x=(-4+-square root(4^2-4*2*(-2))/(2*2)Substitute a = 2, b = -4 and c = -2 into the quadratic formula.
2x=(-4+-square root(16-(-16)))/4Simplify exponents and multiplication inside the parenthesis.
3x=(-4+-square root(32))/4Simplify the subtraction inside the parenthesis.
4x=(-4+-4*square root(2))/4Since 32 = 16·2 and 16·2 = 4√2, rewrite the radical as 4√2.
5x=(-4)/4+-(4*square root(2))/4Use the distributive property of multiplication over addition and subtraction to split the fraction into two fractions.
6x=-1+-square root(2)Simplify each fraction.
7x=-1+square root(2); -1-square root(2)Change the ± into two equations, one using + and the other using -. This quadratic equation has 2 solutions.
Table 9: Solving 2x^2 - 4x - 2 = 0 using the quadratic formula.

Solving a Multiple Variable Equation

When solving a multiple variable equation, one solves for a variable. For example, given the equation x + y = 2, one can solve for the variable x or for the variable y. In this example we will solve for y.

x + y = 2Equation to solve in standard form
x + y - x = 2 - xSubtract x from both sides.
y = 2 - xSimplify the left side of the equation.
y = -x + 2Put equation in slope/intercept form.
Table 10: Solving for y

References

  1. Jones, Burton. Elementary Concepts of Mathematics, pp 141-144. MacMillan and Company, 1947. (Accessed: 2010-01-12). http://www.archive.org/stream/elementaryconcep029487mbp#page/n162/mode/1up/search/equations.

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Solving an Equation. 2008-07-25. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/s/solvingequation.html.

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2008-07-25: Initial version (McAdams, David.)

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