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2.2 Linear equations in one variable  (Page 16/15)

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In this section you will:
  • Solve equations in one variable algebraically.
  • Solve a rational equation.
  • Find a linear equation.
  • Given the equations of two lines, determine whether their graphs are parallel or perpendicular.
  • Write the equation of a line parallel or perpendicular to a given line.

Caroline is a full-time college student planning a spring break vacation. To earn enough money for the trip, she has taken a part-time job at the local bank that pays $15.00/hr, and she opened a savings account with an initial deposit of $400 on January 15. She arranged for direct deposit of her payroll checks. If spring break begins March 20 and the trip will cost approximately $2,500, how many hours will she have to work to earn enough to pay for her vacation? If she can only work 4 hours per day, how many days per week will she have to work? How many weeks will it take? In this section, we will investigate problems like this and others, which generate graphs like the line in [link] .

Coordinate plane where the x-axis ranges from 0 to 200 in intervals of 20 and the y-axis ranges from 0 to 3,000 in intervals of 500. The x-axis is labeled Hours Worked and the y-axis is labeled Savings Account Balance. A linear function is plotted with a y-intercept of 400 with a slope of 15. A dotted horizontal line extends from the point (0,2500).

Solving linear equations in one variable

A linear equation    is an equation of a straight line, written in one variable. The only power of the variable is 1. Linear equations in one variable may take the form   a x + b = 0   and are solved using basic algebraic operations.

We begin by classifying linear equations in one variable as one of three types: identity, conditional, or inconsistent. An identity equation    is true for all values of the variable. Here is an example of an identity equation.

3 x = 2 x + x

The solution set    consists of all values that make the equation true. For this equation, the solution set is all real numbers because any real number substituted for   x   will make the equation true.

A conditional equation    is true for only some values of the variable. For example, if we are to solve the equation   5 x + 2 = 3 x − 6 , we have the following:

5 x + 2 = 3 x − 6 2 x = −8 x = −4

The solution set consists of one number:   { − 4 } .   It is the only solution and, therefore, we have solved a conditional equation.

An inconsistent equation    results in a false statement. For example, if we are to solve   5 x − 15 = 5 ( x − 4 ) , we have the following:

5 x − 15 = 5 x − 20 5 x − 15 − 5 x = 5 x − 20 − 5 x Subtract  5 x  from both sides . −15 ≠ −20 False statement

Indeed,   −15 ≠   −20.   There is no solution because this is an inconsistent equation.

Solving linear equations in one variable involves the fundamental properties of equality and basic algebraic operations. A brief review of those operations follows.

Linear equation in one variable

A linear equation in one variable can be written in the form

a x + b = 0

where a and b are real numbers,   a ≠ 0.

Given a linear equation in one variable, use algebra to solve it.

The following steps are used to manipulate an equation and isolate the unknown variable, so that the last line reads   x = _________, if x is the unknown. There is no set order, as the steps used depend on what is given:

  1. We may add, subtract, multiply, or divide an equation by a number or an expression as long as we do the same thing to both sides of the equal sign. Note that we cannot divide by zero.
  2. Apply the distributive property as needed:   a ( b + c ) = a b + a c .
  3. Isolate the variable on one side of the equation.
  4. When the variable is multiplied by a coefficient in the final stage, multiply both sides of the equation by the reciprocal of the coefficient.

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?
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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.
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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
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"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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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?
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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