Inverse Operations: the pair of operations

 Key Concepts

1. Like Terms, proportion, variables meaning
2. Solving linear equations by combining like terms.
   1. Addition of equations
   2. Subtraction of equations
3. Solving linear equations with variables on both sides
   1. Fractional coefficients
   2. Decimal coefficients
   3. Negative coefficients
4. Solving multi-step equations
   1. Distributive property
   2. Negative coefficient
5. Solving equations with variables on both sides.
6. Solving equations with no solution or infinitely many solutions.
   1. Inspection method
7. Proportional relationships.
8. Slope and interpreting slope.
9. Y-intercept of a line.
10. Analyzing equation of the form y = mx + c.

Inverse Operations: The inverse operations are the pair of operations in which one operation reverses the operation of the other and vice versa. 

Example: Subtraction is the inverse of addition and vice-versa. 

Multiplication is the inverse of division. 

Like Terms: Like terms are the terms that have the same variable and same degree (power).  

Like terms can be added or subtracted from one another. 

3a and 2a are like terms, because although they have different coefficient numbers, they have the exact same letter “a” in them. 

Some other examples of like terms are: 

Example: 4x, 5x are like terms. 

2x², -5x² are like terms. 

Proportion: Proportion tells the equality of two ratios, and it even gives the relation between two variables. 

Variables: The unknown quantity that we represent with the help of the lower-case English alphabets. Because they vary depending on the situation and problem, that is the reason we call them variables.  

Introduction

In this chapter, we are going to learn to analyze and solve linear equations by different methods.Yes, of course! 

Linear equations are equations of  the first order. These equations are defined for lines in the coordinate system. An equation for a straight line is called a linear equation. The general representation is Ax + By = C. 

Operations of linear equations:  

Addition: 

If x + a = b, then x = b − a 

If a number is added on one side of an equation, we may subtract it on the other side.” 

Subtraction 

If  x  − a  = b,  then  x  =  b + a 

“If a number is subtracted on one side of an equation, we may add it on the other side.” 

Multiplication 

If ax  = b, then x = a/b

“If a number multiplies one side of an equation, we may divide it on the other side.” 

Division 

If   x/a = b, then x = ab 

“If a number divides one side of an equation, we may multiply it on the other side.”   

Combining like terms to solve the equation: 

Combing Like terms to solve addition of equations: 

We can  combine like terms to solve the addition of equations by adding their coefficients. 

Example: 

E.g., 1) Simplify the expression 8k-5k? 

Sol: Since these terms (8k and 5k) are like terms, both have k  raised to the first power. We can subtract them by subtracting their coefficients. 

8k-5k 

= (8-5) k 

=3k 

2.1.3: Combine Like Term with Negative Coefficients to solve Equations. 

E.g., 1) Solve the equation -5.2y-6.2y = -11.4  

Sol: 

To combine the like terms with negative coefficients, we use the rules for adding and subtracting rational numbers. 

-5.2y-6.2y = -11.4 

(-5.2-6.2) y = -11.4 

-11.4y = -11.4 

y  = 1  

Solving Equations with Variables on both sides:  

To solve the equations with variables on both sides:  

We first isolate the variables on one side of the equation and constants on the other side of the equation using the addition or subtraction property of equality. 

Combining the like terms, we form a simple linear equation solvable in one step. 

For example: 

E.g., Solve the linear equation

Sol: 

We simply bring all the variables containing terms on one side by inverse operations and all constants on the other side. Coefficients can be either fraction or decimal or integers (positive or negative). 

Solving Equations with Fractional Coefficients: 

E.g., Solve the equation  

Sol:

 

Solving Equations with Decimal Coefficients 

E.g., Solve the equation 0.3x + 0.9x = 134 ? 

Sol: 

Solving Equations with Negative Coefficients: 

E.g., Solve the equation

Sol: 

Solving Multi-step Equations:  

Multi-step equations are algebraic expressions that require more than one operation, such as subtraction, addition, multiplication, division, or exponentiation, to solve. 

Multi-step involves applying distributive property and then solving the linear equation. 

Application of Distributive Property to solve Multi-Step Equations 

The distributive property of multiplication over addition can be used when you multiply a number by a sum. 

Distributive Property: The distributive property is the name given to the following  process:  

a (b + c) = ab + ac 

As the variable is distributed to the terms in the bracket, we call this property a distributive property. This property holds good for both numerals and variables. 

E.g., Solve the equation 5x+3=18+2(x+9). 

Sol: 

5x+3=18+2(x+9) 

5x+3=18+2x+18                            (∴a (b + c) = ab + ac)) 

5x+3=2x+36 

5x-2x=36-3 

       3x=33 

         X =11 

Application of Negative Coefficient Distributive Property 

E.g., Solve the equation -3(x – 1) + 7x = 27. 

Sol:     -3(x – 1) + 7x = 27. 

-3x -3(-1) + 7x = 27 

-3x + 3 + 7x = 27 

 4x + 3 = 27 

 4x + 3 – 3 = 27 – 3  

 4x = 24 

x = 6 

Using Distributive Property on Both Sides of the Equation 

E.g., Solve the equation  ? 

Sol:

       Given  

       

Combine multiplied terms into a single fraction  

Multiply all terms by the same value to eliminate fraction denominators  

Cancel multiplied terms that are in the denominator, then we get  

   x+4=2x−16 

  Subtract 4 from both sides of the equation  

   x+4−4=2x−16−4 

   x=2x−20

   Subtract 2x from both sides of the equation  

    x−2x=2x−20−2x 

   −x=−20 

   ∴ x=20 

Equations with no solution or infinitely many solutions:  

Solving equation having Infinitely Many Solutions:  

Infinitely many solutions: The equations like 5x = 5x are true for any real value of x . So, such types of equations will have infinitely many solutions.  Equations that are for any real value of the variable in the equation, such equations would have infinitely many solutions. 

E.g.,  Determine the number of solutions of 7(8x+5) −35=4(14x)? 

Sol: 

Expand the equation 7(8x+5) −35=4(14x) 

   56x+35-35 = 56x 

      56x = 56x  

The equation having infinitely many solutions. 

Solving equation having One solution: 

A one-variable equation has a one-solution, when solving results in one value for the variable. 

E.g., Determine the number of solutions of 4(x+2) = x+8+56? 

Sol: 

4(x+2) = x+8+56 

     4x+8 = x+64 

         3x = 56 

          X  = 18.6 

The equation 4(x+2) = x+8+56 has only o ne solution. 

Solving equation having No Solution. 

Equations that are not true for any real value of the variable in the equation, such equations would have no solution.  

Equations like x+5 = x+6  will have no solution. 

E.g.,  Determine the number of solutions of 3(2x+5) = 2(3x+3)? 

Sol:  

Linear equations may

Identifying the Number of Solutions by Inspection Method 

Determine the number of solutions that the equations have by inspection method? 

E.g., 

a)  x + 5 = 2x-x-15+10 

Sol: 

x + 5 = x-5  

The equivalent expressions are not true for any value of x. The equation has no solution. 

b)  7(8x+5) −35=4(14x) 

Sol:

56x+35-35 = 56x 

      56x= 56x  

The equivalent equation is true for any value of x. The equation has infinitely many solutions. 

Comparing Proportional relationships:  

Comparing Proportional Relationships represented in Tables and Graphs. 

The comparison is done between two like items but, here instead of comparing two tables or two graphs, we compare a table to a graph. This is done by taking the tabular values to unit rate and graphical value to a unit rate. These unit rates are compared to find the greater value and fewer value functions. 

Example: Manisha is researching the cruising speeds of different planes. Which airplane has a greater cruising speed? 

Sol: 

Find the cruising speed of the Cessna.  

The Cessna has a cruising speed of 8 kilometers per minute. 

Find the cruising speed of Boeing 747. 

The Boeing 747 has a cruising speed of 15 kilometers per minute. 

So, The Boeing 747 has a greater cruising speed than the Cessna. 

Example: John observed the distance covered by the train at different times in a tabular form and graphed the time vs distance graph for the bus. Observe the graph and table and find which transport is faster? 

Solution: Clearly, we cannot directly say the transport with greater speed. Coming to table  we find the speed for unit rate. 

For every 10 min, it covers 12 km 

For every 1 min, it covers

1210=1.2 km 12/10=1.2 km

Now, checking the graph, we see that for 1 unit x-value, the graph moves to 2 units of the y-axis. 

1 unit of x-value = 2 units of y-value 

Now, both graph and table unit rates are known. 

By comparing, we can say that 2 > 1.2.  

So, the speed of the bus is greater than the train.  

Comparing Proportional Relationships represented in Graphs and Equations: 

In this section we instead of comparing graphs to graphs or equations to equations, we compare graphs to equations. We find the unit rate from the graph; we substitute the value of x = 1 in the equation to get the unit rate of the equation. We compare both the unit rates and get the function with a greater increase.  

Example: 

Student ‘A’ was given a graph and student ‘B’ was given an equation y = 2x.  The task was both the students need to discuss and answer which has the best unit rate? 

Solution: The unit rate of graph of student A is  

For 1 unit  of x-value the graph moves to 1 unit of y-value 

1 unit x-value = 1 unit y-value 

 it rate = 1 

The unit rate of the equation of student B is 

Substitute  the value of x = 1 in the equation. The y-value we get is the unit rate of the equation. 

y = 2x 

y = 2(1) 

y = 2  

Unit rate = 2 

Clearly, we can say that the equation has a better unit rate than the graph given. 

Comparing Proportional Relationships represented in Graphs and Verbal Description 

In this section we instead of comparing a graph to other, we compare a graph to a statement. This is done by taking the unit rate. We find the unit rate of the graph and we will find the unit rate of the description of the statement. We compare both the unit rate to find the greater one and the lesser one. 

Example: The graph represents the weight vs cost of the iron. 15 kg aluminium costs $150. 

Solution:  

From the graph for every 5 units of x-value, the graph moves to 50 units y-value. 

Every 1 unit of x-value units of y-value. 

The unit rate of the graph is 10. 

The description is ‘the cost of 15kg aluminium is $150.’ 

1 kg costs

The unit rate is 10.  

Since both the unit rates are equal, both represents the same function or equation. 

Proportional relationships and Slopes:  

Slope: 

Slope: The slope of the line is given as the ratio between the vertical rise and the horizontal run. The slope is generally denoted by m. 

Rise:  The change in vertical distance is called rise. 

Run: The change in horizontal distance is called run. 

E.g.,  Calculate the slope of the line going through the point (1, 2) and the point (4, 3)? 

Sol: 

Slope = Rise/Run 

Slope = Change in Y/Change in X 

Slope = 3-2/4-1 

Slope = 1/3  

To Find the Slope from Two Points 

When two distinct points on the line are given (x₁, y₁) and ( x₂, y₂), then the slope is given by   

• Keep  (x, y) are variables. 

• (x₁, y₁) and (x₂, y₂)  are two points on the line.   

E.g., Calculate the slope of a line passing through the points (3, 2) and (5, 3)? 

Sol: 

The slope of a line is the proportional constant. 

Interpreting Slope: 

E.g., The graph shows the distance a car traveling over time. Find the slope of the line? 

Sol: 

  Slope =  rise / run 

Linear Equation (y = mx):  

Relating Constant of Proportionality to Slope 

Constant of proportionality:  

y =mx, where m is the constant of proportionality, ‘y’ is the dependent quantity and ‘x’ is the independent quantity. 

In proportional relationships, the constant proportional ‘m’ is equal to the slope. 

E.g., Find the constant proportionality if, y = 24 and x = 3 and y  x? 

Sol: 

We know that the equation of proportionality is y =mx.  

Substitute the values of x and y in the above equation gives m value. 

24 = m3 

m = 8. 

To Write the Linear Equation from two Points: 

If the highest power of the variables is 1 in an equation, then the equation is a linear equation. Graphically all linear equations are straight lines with a slope equal to the constant of proportionality of the two variables. 

When two distinct points on the line are given, with this equation, we can find the linear equation. 

E.g., Find the equation of the straight line passing through the points (2, 3) and (6, -5)? 

Sol: 

The equation of a line passing through two points (x₁, y₁) and (x₂, y₂) is 

Understand y-intercept of a line 

Determine the y-intercept of a relationship. 

The y-intercept of line: The value of y when x is substituted to 0 is called as the y-intercept. 

Graphically, the y-intercept is the value where the line cuts the y-axis. 

E.g., Find the y-intercept of the line y = 3x-1? 

Sol: 

To find the y-intercept set x = 0, then  

          y = 3(0)-1 

          y = -1 

The x-intercept of line: The value of x when y is substituted to 0 is called as x-intercept. 

Graphically, the x-intercept is the value where the line cuts the x-axis. 

Note: Intercepts are not lengths.  

The y-intercept of a Proportional Relationship: 

Graphing a proportional relationship will always have a y-intercept through the origin (0, 0). The unit rate will become the slope of the graph as the unit rate is the same as the slope of the line. 

E.g., 1) A manufacturer company manufactures a set of the number of parts per minute as shown in the graph below. Find how many parts that the company manufactured when it is first turned on? 

Sol: 

The company has not made any parts when it is first turned on. So the answer should be zero.  

Identifying the y-Intercept: 

E.g., 1) What is the y-intercept for the linear relationship shown below?  

Sol: 

The line crosses the y-axis at (0, 2). The y-intercept is 2. 

E.g., 2) What is the y-intercept for the linear relationship shown below? 

Sol: The line crosses the y-axis at (0, -1). The y-intercept is -1. 

Analyzing equation of the form y = mx + c 

Write the Equation of the Line: 

The general linear equation is given by y = mx + c. The equation in slope-intercept form, where ‘m’ is the slope of the line and ‘c’ is the y-intercept. 

y = mx + c 

y = c + xm 

Variables: 

x and y 

Constants: 

m and c 

On graph: 

m = gradient 

c = y-axis intercept 

x and y have a linear relationship 

E.g., 

Find the slope of the equation y = 18x + 5? 

Sol: 

The equation is in slope intercept form y = mx + c. 

The slope (m) = 18 

And y-intercept = 5. 

Write a Linear Equation Given its Graph: 

E.g., 

Identify the slope and y-intercept from the graph and find the equation of the line? 

Sol: 

              Slope = 

            From the graph y-intercept = -1 

The general linear equation is given by y = mx + c. 

Graph a Linear Equation: 

E.g., Graph the equation ?  

Sol:  The y-intercept is 5. Plot the point at (0, 5). 

The slope of the line is -1/2 . Another point on the line is (6, 2).  

Exercise :

1. Solve for x: 4/9 x+1/5 x=87  ?
2. Solve for x:  x+0.15x=3.45
3. Solve the equation -2/5 x+3=2/3 x+1/3  ?
4. Solve the equation 1/6(x-5)=1/2(x+6) ?
5. What is the slope of the line?
6. Find the y-intercept of the graph?
   • 3/4 x+x-5=10+2x
   • b) 3x-2.7=2x+2.7+x
   • c) 9x+4.5-2x=2.3+7x+2.2
   • d) 1/5 x-7=3/4+2x-25 3/4
7. What is the slope of the line?

8. Find the y-intercept of the graph

9. Graph the equation y = 3x – 5?
10. Write an equation for the line in slope-intercept form?

What Have We Learned

• Understand the definition of like terms, inverse operation, proportion, and variation.
• How to combine the like terms to solve the equation.
• How to solve the equation with variables on both sides.
  • 550-25x=10+15x+20x    
• How to solve multi-step equations. (Distributive property)
• Understand equations with no solution, one solution and infinitely many solutions. (inspection method).
• Understand how to compare proportional relationships. (tables, graphs, verbal description, and equations).
• Understand the definition of rise, run, slope and interpreting slope. ( Slope=Rise/Run  ).
• Understand how to find slope from two points.
• Understand the definition of constant proportionality and how to find the constant proportionality.
• Understand how to find a linear equation from two points.
• Understand how to identify and determine the y-intercept of a relationship.
• Understand analyzing equation of the form y = mx + c.
• Understand how to graph a linear equation.

Concept Map: