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Welcome to our Math lesson on Equations for solving Word Problems, this is the fifth lesson of our suite of math lessons covering the topic of Word Problems Involving Equations, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
There are many types of equations besides the first - order equations with one variable. We will discuss a few examples where two other types of equations (second - order equations with one variable and first - order equations with two variables) are used in word problems. Let's present a short overview of these two types of equations.
Second - order equations with one variable include equations that contain a single variable raised at the second power. In the next tutorials, we will see that the general form of these equations (otherwise known as 'quadratic equations') is
The left part of such equations is identical to the general form of quadratic expressions discussed in tutorial 6.4, where we explained how to factorise such expressions.
In this paragraph, however, we will consider only a simplified form of quadratic equations, where the coefficient b is 0. Therefore, the problems discussed here will require the use of the reduced quadratic equation
Let's consider an example to clarify this point.
The area of a square is 4 cm2 more than that of a rectangle of dimensions 9 cm × 5 cm. What is the side length of the square?
The unknown in this problem is the side length of the square, which we denote by x. Since the area of a rectangle is the product of its dimensions and that of a square is the second power of the side length, we obtain the equation
Hence, the side length of the square is
(√49 can be 7 or - 7. However, we consider only the positive root, as the side length cannot be negative.)
First - order equations with two variables have the general form
where a and b are coefficients, c is a constant, while x and y are the two variables. For example, 3x - 2y - 6 = 0 is a first - order equation with two variables as it contains two variables x and y where both are in the first power (hence the name "first - order equation with two variables").
To solve first - order equations with two variables means finding the values of the two variables. However, since there are a lot of possible combinations that can give a true result, we must have some additional information about the relationship between the two variables. For example, in our example, we may take x = 4 and y = 3 as possible solutions, as 3 · 4 - 2 · 3 - 6 = 0. Yet, we may also take x = 2 and y = 0 as possible solutions, as 3 · 2 - 2 · 0 - 6 = 0. Both pairs of solutions are true. However, in a word problem, usually, there are only a couple of values that fit the description. Therefore, as said earlier, we must have some extra information about the relationship between the two variables. This extra information usually forms another first - order equation and we solve it as a system (we will explain systems of equations in the last two tutorials of this chapter). However, for now, it is sufficient to know that we express one of the variables in terms of the other variable and in this way, the original equation turns into a first - order equation with one variable, which we already know how to solve.
For example, if we have given that in a class there are 30 students where the boys vs girls ratio is B:G = 2:3, we can write
This is a first - order equation (variables are written as B and G) with two variables, as we can write it as
or
As we said earlier, it is impossible to find the correct value of the two variables only through this equation, as there are many combinations that fit the description. Therefore, we use the help of the other clue, which says that the total number of students in the class is 30. Hence, we have the other equation
available, which helps to express one of the variables (for example the girls G) in terms of the other variable in the first equation. Therefore, since G = 30 - B, we can substitute this expression in the first equation to obtain
Now, we have a first - order equation with one variable, which can be easily solved. Thus,
In this first - order equation with one variable we have a = 5 and b = -60. Therefore, the root B that gives the number of boys in the class is
Hence, the number of girls in the class is
The perimeter of a rectangle is 12 cm more than that of an equilateral triangle with a 16 cm side length. Calculate the dimensions of the rectangle if its length is 2 cm more than its width.
We can express the rectangle's length by x and its width by y. Since the perimeter (the sum of all sides length) of a rectangle is calculated by the formula P = x + x + y + y = 2x + 2y, and given that an equilateral triangle has three equal sides, we have
This is a first - order equation with two variables. Therefore, we need some additional information to solve it. This additional information is already available. We know that length is 2 cm more than width, i.e. x = y + 2. Therefore, we substitute x with its equivalent expression in the main equation to obtain
In this way, we obtained a first - order equation with one variable, where a = 2 and b = -28. Therefore, we obtain for the variable y:
Hence, since x + y = 30, we have
This too is a first - order equation with one variable, where a = 1 and b = -16. Therefore, we obtain for the other variable x:
Hence, the length of the rectangle is 16 cm and the width is 14 cm.
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