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x + 1 = 10

conditional, x = 9

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y 4 = 7

conditional, y = 11

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5 a = 25

conditional, a = 5

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x 4 = 9

conditional, x = 36

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18 b = 6

conditional, b = 3

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y 2 = y 2

identity

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x + 4 = x 3

contradiction

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x + x + x = 3 x

identity

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8 x = 0

conditional, x = 0

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m 7 = 5

conditional, m = 2

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Literal equations

Literal equations

Some equations involve more than one variable. Such equations are called literal equations .

An equation is solved for a particular variable if that variable alone equals an expression that does not contain that particular variable.

    The following equations are examples of literal equations.

  1. y = 2 x + 7 . It is solved for y .
  2. d = r t . It is solved for d .
  3. I = p r t . It is solved for I .
  4. z = x u s . It is solved for z .
  5. y + 1 = x + 4 . This equation is not solved for any particular variable since no variable is isolated.

Solving equation of the form x + a = b and x a = b

Recall that the equal sign of an equation indicates that the number represented by the expression on the left side is the same as the number represented by the expression on the right side.

This is the this number same as number x = 6 x + 2 = 8 x 1 = 5

    This suggests the following procedures:

  1. We can obtain an equivalent equation (an equation having the same solutions as the original equation) by adding the same number to both sides of the equation.
  2. We can obtain an equivalent equation by subtracting the same number from both sides of the equation.

We can use these results to isolate x , thus solving for x .

Solving x + a = b For x

x + a = b The a is associated with x by addition . Undo the association x + a a = b a by subtracting a from b o t h sides . x + 0 = b a a a = 0 and 0 is the additive identity . x + 0 = x . x = b a This equation is equivalent to the first equation, and it is solved for x .

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Solving x a = b For x

x a = b The a is associated with x by subtraction . Undo the association x a + a = b + a by adding a to b o t h sides . x + 0 = b + a a + a = 0 and 0 is the additive identity . x + 0 = x . x = b + a This equation is equivalent to the first equation, and it is solved for x .

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Method for solving x + a = b And x a = b For x

To solve the equation x + a = b for x , subtract a from both sides of the equation.
To solve the equation x a = b for x , add a to both sides of the equation.

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Sample set b

Solve x + 7 = 10 for x .

x + 7 = 10 7 is associated with x by addition . Undo the association x + 7 7 = 10 7 by subtracting 7 from b o t h sides . x + 0 = 3 7 7 = 0 and 0 is the additive identity . x + 0 = x . x = 3 x is isolated, and the equation x = 3 is equivalent to the original equation x + 7 = 10. Therefore, these two equation have the same solution . The solution to x = 3 is clearly 3. Thus, the solution to x + 7 = 10 is also 3.

Check : Substitute 3 for x in the original equation. x + 7 = 10 3 + 7 = 10 Is this correct? 10 = 10 Yes, this is correct .

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Solve m 2 = 9 for m .

m 2 = 9 2 is associated with m by subtraction . Undo the association m 2 + 2 = 9 + 2 by adding 2 from b o t h sides . m + 0 = 7 2 + 2 = 0 and 0 is the additive identity . m + 0 = m . m = 7

Check : Substitute 7 for m in the original equation. m 2 = 9 7 2 = 9 Is this correct? 9 = 9 Yes, this is correct .

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Use a calculator to solve this equation. Solve y 2.181 = 16.915 for y .

y 2.181 = 16.915 y 2.181 + 2.181 = 16.915 + 2.181 y = 14.734

On the Calculator
Type 16.915 Press + / Press + Type 2.181 Press = Display reads: 14.734

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Solve y + m = s for y .

y + m = s m is associated with y by addition . Undo the association y + m m = s m by subtracting m from b o t h sides . y + 0 = s m m m = 0 and 0 is the additive identity . y + 0 = y . y = s m

Check : Substitute s m for y in the original equation. y + m = s s m + m = s Is this correct? s = s True Yes, this is correct .

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Solve k 3 h = 8 h + 5 for k .

k 3 h = 8 h + 5 3 h is associated with k by subtraction . Undo the association k 3 h + 3 h = 8 h + 5 + 3 h by adding 3 h to b o t h sides . k + 0 = 5 h + 5 3 h + 3 h = 0 and 0 is the additive identity . k + 0 = k . k = 5 h + 5

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Practice set b

Solve y 3 = 8 for y .

y = 11

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Solve x + 9 = 4 for x .

x = 13

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Solve m + 6 = 0 for m .

m = 6

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Solve g 7.2 = 1.3 for g .

g = 8.5

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solve f + 2 d = 5 d for f .

f = 3 d

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Solve x + 8 y = 2 y 1 for x .

x = 6 y 1

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Solve y + 4 x 1 = 5 x + 8 for y .

y = x + 9

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Exercises

For the following problems, classify each of the equations as an identity, contradiction, or conditional equation.

g + g + g + g = 4 g

identity

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For the following problems, determine which of the literal equations have been solved for a variable. Write "solved" or "not solved."

4 a = y 6

not solved

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For the following problems, solve each of the conditional equations.

y + 6 = 11

y = 17

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g + 164 = 123

g = 287

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x + 17 = 426

x = 443

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y + 17.003 = 1.056

y = 18.059

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Solve n + m = 4 for n .

n = 4 m

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Solve P + 3 Q 8 = 0 for P .

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Solve a + b 3 c = d 2 f for b .

b = a + 3 c + d 2 f

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Solve x 3 y + 5 z + 1 = 2 y 7 z + 8 for x .

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Solve 4 a 2 b + c + 11 = 6 a 5 b for c .

c = 2 a 3 b 11

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Exercises for review

( [link] ) Simplify ( 4 x 5 y 2 ) 3 .

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( [link] ) Write 20 x 3 y 7 5 x 5 y 3 so that only positive exponents appear.

4 y 4 x 2

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( [link] ) Write the number of terms that appear in the expression 5 x 2 + 2 x 6 + ( a + b ) , and then list them.

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( [link] ) Find the product. ( 3 x 1 ) 2 .

9 x 2 6 x + 1

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( [link] ) Specify the domain of the equation y = 5 x 2 .

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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
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Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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