Writing Equations with Tables, Graphs, and Word Problems - Amped ... - Free Printable

Problem Analysis:

The task involves writing equations based on given tables and graphs. Let's solve each part step by step.

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Part 1: Use the table to write two equations

#### Problem 1:
| $x$ | $y$ |
|------|-----|
| 1 | 4 |
| 2 | 8 |
| 3 | 12 |

- Step 1: Identify the relationship between $x$ and $y$.
- When $x = 1$, $y = 4$.
- When $x = 2$, $y = 8$.
- When $x = 3$, $y = 12$.
- The pattern shows that $y = 4x$.

- Step 2: Write the equations.
- $x = \frac{y}{4}$
- $y = 4x$

#### Answer for Problem 1:
$$
x = \frac{y}{4}, \quad y = 4x
$$

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#### Problem 2:
| $x$ | $y$ |
|------|-----|
| 21 | 3 |
| 28 | 4 |
| 35 | 5 |

- Step 1: Identify the relationship between $x$ and $y$.
- When $x = 21$, $y = 3$.
- When $x = 28$, $y = 4$.
- When $x = 35$, $y = 5$.
- The pattern shows that $y = \frac{x}{7}$.

- Step 2: Write the equations.
- $x = 7y$
- $y = \frac{x}{7}$

#### Answer for Problem 2:
$$
x = 7y, \quad y = \frac{x}{7}
$$

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#### Problem 3:
| $x$ | $y$ |
|------|-----|
| -2 | 4 |
| 0 | 6 |
| 2 | 8 |

- Step 1: Identify the relationship between $x$ and $y$.
- When $x = -2$, $y = 4$.
- When $x = 0$, $y = 6$.
- When $x = 2$, $y = 8$.
- The pattern shows that $y = x + 6$.

- Step 2: Write the equations.
- $x = y - 6$
- $y = x + 6$

#### Answer for Problem 3:
$$
x = y - 6, \quad y = x + 6
$$

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#### Problem 4:
| $x$ | $y$ |
|------|-----|
| 34 | 19 |
| 40 | 25 |
| 46 | 31 |

- Step 1: Identify the relationship between $x$ and $y$.
- When $x = 34$, $y = 19$.
- When $x = 40$, $y = 25$.
- When $x = 46$, $y = 31$.
- The pattern shows that $y = x - 15$.

- Step 2: Write the equations.
- $x = y + 15$
- $y = x - 15$

#### Answer for Problem 4:
$$
x = y + 15, \quad y = x - 15
$$

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Part 2: Use the graph to write an equation in terms of $y$

#### Problem 5:
- Graph Analysis: The points are approximately:
- $(2, 60)$
- $(3, 80)$
- $(4, 100)$

- Step 1: Identify the relationship.
- The $y$-values increase by 20 as $x$ increases by 1.
- The equation is linear: $y = 20x + 20$.

#### Answer for Problem 5:
$$
y = 20x + 20
$$

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#### Problem 6:
- Graph Analysis: The points are approximately:
- $(30, 20)$
- $(60, 40)$
- $(90, 60)$

- Step 1: Identify the relationship.
- The $y$-values increase by 20 as $x$ increases by 30.
- The slope is $\frac{20}{30} = \frac{2}{3}$.
- Using the point $(30, 20)$, the equation is $y = \frac{2}{3}x$.

#### Answer for Problem 6:
$$
y = \frac{2}{3}x
$$

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#### Problem 7:
- Graph Analysis: The points are approximately:
- $(10, 2)$
- $(15, 3)$
- $(20, 4)$

- Step 1: Identify the relationship.
- The $y$-values increase by 1 as $x$ increases by 5.
- The slope is $\frac{1}{5}$.
- Using the point $(10, 2)$, the equation is $y = \frac{1}{5}x$.

#### Answer for Problem 7:
$$
y = \frac{1}{5}x
$$

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Final Answers:

1. $x = \frac{y}{4}, \quad y = 4x$
2. $x = 7y, \quad y = \frac{x}{7}$
3. $x = y - 6, \quad y = x + 6$
4. $x = y + 15, \quad y = x - 15$
5. $y = 20x + 20$
6. $y = \frac{2}{3}x$
7. $y = \frac{1}{5}x$

$$
\boxed{
\begin{aligned}
&1. \, x = \frac{y}{4}, \, y = 4x \\
&2. \, x = 7y, \, y = \frac{x}{7} \\
&3. \, x = y - 6, \, y = x + 6 \\
&4. \, x = y + 15, \, y = x - 15 \\
&5. \, y = 20x + 20 \\
&6. \, y = \frac{2}{3}x \\
&7. \, y = \frac{1}{5}x
\end{aligned}
}
$$