Question

Use the graph to complete each statement.
Use the point (−1,−2) as the starting point.
Enter your answers in the boxes.
Coordinate grid showing two points. The ordered pairs for the points are 5 comma 1 and negative 1 comma negative 2. A line goes through the two points.

The horizontal change between the points is ___.

The vertical change between the points is ___.

Answer 1

To find the horizontal and vertical changes between the points \((-1, -2)\) and \((5, 1)\), we can use the following formulas:

1. **Horizontal Change (Δx)**:
\[
\Delta x = x_2 - x_1
\]
where \((x_1, y_1) = (-1, -2)\) and \((x_2, y_2) = (5, 1)\).

Substituting the values:
\[
\Delta x = 5 - (-1) = 5 + 1 = 6
\]

2. **Vertical Change (Δy)**:
\[
\Delta y = y_2 - y_1
\]
Substituting the values:
\[
\Delta y = 1 - (-2) = 1 + 2 = 3
\]

Now, we can complete the statements:

The horizontal change between the points is **6**.

The vertical change between the points is **3**.

Answer 2

What is the slope of the line?

A. −3

B. −1/2

C. 1/2

D. 2

Answer 3

To find the slope of the line that passes through the points \((−1, −2)\) and \((5, 1)\), we can use the formula for the slope \(m\):

\[
m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substituting in the values from the points:

- \(y_1 = -2\)
- \(y_2 = 1\)
- \(x_1 = -1\)
- \(x_2 = 5\)

Calculating the differences:

\[
\Delta y = 1 - (-2) = 1 + 2 = 3
\]
\[
\Delta x = 5 - (-1) = 5 + 1 = 6
\]

Now substituting these values into the slope formula:

\[
m = \frac{3}{6} = \frac{1}{2}
\]

The slope of the line is **C. \( \frac{1}{2} \)**.

Answer 4

Bronze is a combination of copper and tin. A bronze tool with a 0.5 kg handle will have x kg of copper and y kg of tin with this equation: x−7y=0.5.
Can the tool have 3 kg of copper and 0.6 kg of tin?

First, let's substitute the appropriate numbers for the variables into our equation x−7y=0.5.

__ −7(__) = 0.5

Answer 5

To determine if the bronze tool can have 3 kg of copper and 0.6 kg of tin, we will substitute the values into the given equation \(x - 7y = 0.5\).

Here, \(x = 3\) (the amount of copper) and \(y = 0.6\) (the amount of tin).

The equation becomes:

\[
3 - 7(0.6) = 0.5
\]

Now calculate \(7 \times 0.6\):

\[
7 \times 0.6 = 4.2
\]

Now substitute this value back into the equation:

\[
3 - 4.2 = 0.5
\]

Calculating the left side:

\[
3 - 4.2 = -1.2
\]

Now we compare this result to the right side of the equation:

\[
-1.2 \neq 0.5
\]

Thus, the substitution does not satisfy the equation. Therefore, **the tool cannot have 3 kg of copper and 0.6 kg of tin**.

The final expression to fill in is:

**3 − 7(0.6) = 0.5**.

Answer 6

Simplify the left side of the equation.

__ = 0.5

Answer 7

We will simplify the left side of the equation using the values we previously substituted:

We have:

\[
3 - 7(0.6)
\]

Now let's calculate \(7(0.6)\):

\[
7 \times 0.6 = 4.2
\]

Then, we substitute this value back into the equation:

\[
3 - 4.2
\]

Now, perform the subtraction:

\[
3 - 4.2 = -1.2
\]

So, when we write it out, we simplify the left side as:

\[
-1.2 = 0.5
\]

To clarify:

The simplified left side of the equation is **-1.2**. Therefore, we can write:

\[
-1.2 = 0.5
\]

This shows that the left side does not equal 0.5, confirming earlier that the tool cannot have 3 kg of copper and 0.6 kg of tin.