Learn on PengienVision, Mathematics, Grade 6Chapter 4: Represent and Solve Equations and Inequalities

Lesson 9: Use Patterns to Write and Solve Equations

Property A relationship between two quantities, or variables, can be represented in different ways. A table shows specific pairs of values that follow a rule, while an equation describes that same rule algebraically.

Section 1

Representing Relationships: Tables and Equations

Property

A relationship between two quantities, or variables, can be represented in different ways. A table shows specific pairs of values that follow a rule, while an equation describes that same rule algebraically.

Examples

Table: A table can show how the cost (yy) relates to the number of tickets purchased (xx).

Tickets (x)Cost (y)1$52$103$15\begin{array}{|c|c|} \hline \textbf{Tickets (x)} & \textbf{Cost (y)} \\ \hline 1 & \$5 \\ \hline 2 & \$10 \\ \hline 3 & \$15 \\ \hline \end{array}

Equation: The same relationship can be represented by the equation:

y=5xy = 5x

Table: A table can show how the total cost (yy) relates to the number of notebooks purchased (xx).

Notebooks (x)Total Cost (y)1$52$73$9\begin{array}{|c|c|} \hline \textbf{Notebooks (x)} & \textbf{Total Cost (y)} \\ \hline 1 & \$5 \\ \hline 2 & \$7 \\ \hline 3 & \$9 \\ \hline \end{array}

Equation:

  1. Look at how the total cost changes as xx increases by 1:
7−5=2,9−7=2 7 - 5 = 2, \quad 9 - 7 = 2

→ Each notebook adds $2.

  1. Notice that when x=1x = 1, y=5y = 5.
  2. Combine these observations:
y=5+2(x−1)→y=3+2xy = 5 + 2(x - 1) → y = 3 + 2x

Explanation

A table and an equation can describe the same mathematical relationship. The table lists specific examples of the relationship, while the equation provides a general rule that works for any pair of values. In this lesson, you will learn how to find the pattern in a table to write its corresponding equation.

Lesson overview

Expand to review the lesson summary and core properties.

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Section 1

Representing Relationships: Tables and Equations

Property

A relationship between two quantities, or variables, can be represented in different ways. A table shows specific pairs of values that follow a rule, while an equation describes that same rule algebraically.

Examples

Table: A table can show how the cost (yy) relates to the number of tickets purchased (xx).

Tickets (x)Cost (y)1$52$103$15\begin{array}{|c|c|} \hline \textbf{Tickets (x)} & \textbf{Cost (y)} \\ \hline 1 & \$5 \\ \hline 2 & \$10 \\ \hline 3 & \$15 \\ \hline \end{array}

Equation: The same relationship can be represented by the equation:

y=5xy = 5x

Table: A table can show how the total cost (yy) relates to the number of notebooks purchased (xx).

Notebooks (x)Total Cost (y)1$52$73$9\begin{array}{|c|c|} \hline \textbf{Notebooks (x)} & \textbf{Total Cost (y)} \\ \hline 1 & \$5 \\ \hline 2 & \$7 \\ \hline 3 & \$9 \\ \hline \end{array}

Equation:

  1. Look at how the total cost changes as xx increases by 1:
7−5=2,9−7=2 7 - 5 = 2, \quad 9 - 7 = 2

→ Each notebook adds $2.

  1. Notice that when x=1x = 1, y=5y = 5.
  2. Combine these observations:
y=5+2(x−1)→y=3+2xy = 5 + 2(x - 1) → y = 3 + 2x

Explanation

A table and an equation can describe the same mathematical relationship. The table lists specific examples of the relationship, while the equation provides a general rule that works for any pair of values. In this lesson, you will learn how to find the pattern in a table to write its corresponding equation.