14.3 Solving linear equations using multiplication and division: xa+b=c\frac{x}{a} + b = c

You plan to visit your grandmother who lives in the other side of the town but you have to make it back before dinner. If you drive for 2 and a half hours at an average speed of 40 miles per hour going back home how far is your grandmother’s house?

This problem can be easily solved through linear equations. Linear equations show the increase or decrease of a number in a particular pattern. A Linear Equation has generally the form, y= mx + b where m is the slope, and b is the y intercept. A linear equation has two variables x and y, and using this equation could help you solve the distance of your grandmother’s house to your house.

In section 1, we will be learning all about modeling and solving one step linear equations. These equations have the general form of ax =b, = b. Examples of one step linear equations are 3x=6, -4x=56 and in order for you to solve for x you just need to divide both sides of the equation with a, in the case of our examples, a is 3 and -4 respectively.

Linear equations can also be added and subtracted when they come in the ax + b = c. C stands for constant, and this number will be transposed to the other side of the equation in order to solve for x, like say 3x + 9 = 21, it would become 3x = 21-9 and so x is 4. Notice that 9 uses the opposite operation when transposed to the other side of the equation, this stays true for all the other numbers that you will be transposing. We will get to learn on how to solve more on these kinds of problems and also learn how to translate diagrams illustrating addition and subtraction of linear equations in section 2. While in section 3 we will learn how to multiply and divide linear equations.

For the last part of this chapter, we will learn how to solve linear equations a (x + b) = c using distributive property. As illustrated by the given equation a (x+b) = c, a is distributed to x and b via multiplication. Using this property, we will be able to solve for x.

It would be very handy if we are very familiar with integers and the linear relations so feel free to go back to these chapters if you need to.

Solving linear equations using multiplication and division: xa+b=c\frac{x}{a} + b = c

Continue from the previous section, we will solve linear equations in diagram, equation, and word problems forms. However, in addition to subtraction and addition, we will teach you how to solve linear equations that also involve multiplications and divisions.


  • 1.
    Find x in the equation modelled by each diagram.
  • 2.
  • 3.
    The amount of time needed for an ice cube to melt when the temperature is between 10 °C to 30 °C can be expressed by the equation t=16C2t = 16 - \frac{C}{2}, where t is the amount of time for the ice cube to melt in minute; and C is the temperature in degree Celsius.
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Solving two-step linear equations using multiplication and division: xa+b=c\frac{x}{a} + b = c

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