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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 ___.

7 answers

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.

What is the slope of the line?

A. −3

B. −1/2

C. 1/2

D. 2

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. \( \frac{1}{2} \).

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

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.

Simplify the left side of the equation.

__ = 0.5

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.