Math Problem Statement

For the equation y = -4x - 5, find the y-coordinate for each value of x (0, 1, 2, 3). Also, find the initial value when x = 0 and determine how much the y-value changes for each 1-unit increase in the x-value.

Solution

The equation given is y=−4x−5y = -4x - 5. We need to find the yy-coordinate for each value of xx, and answer the following questions.

Step 1: Finding the yy-coordinates

We will substitute the values of xx (0, 1, 2, 3) into the equation to calculate the corresponding yy-values.

  • For x=0x = 0: y=−4(0)−5=−5y = -4(0) - 5 = -5 So, y=−5y = -5.

  • For x=1x = 1: y=−4(1)−5=−4−5=−9y = -4(1) - 5 = -4 - 5 = -9 So, y=−9y = -9.

  • For x=2x = 2: y=−4(2)−5=−8−5=−13y = -4(2) - 5 = -8 - 5 = -13 So, y=−13y = -13.

  • For x=3x = 3: y=−4(3)−5=−12−5=−17y = -4(3) - 5 = -12 - 5 = -17 So, y=−17y = -17.

Step 2: Filling the Table

The table with xx- and yy-coordinates is:

\hline x & y \\ \hline 0 & -5 \\ 1 & -9 \\ 2 & -13 \\ 3 & -17 \\ \hline \end{array}$$ ### Step 3: Initial Function Value The initial value of the function occurs when $$x = 0$$. From the calculations above, when $$x = 0$$, $$y = -5$$. So, the function's initial value is $$-5$$. ### Step 4: Change in $$y$$-value per Unit Increase in $$x$$ The change in $$y$$-value for each 1-unit increase in $$x$$ is the slope of the equation, which is $$-4$$. This means that for every increase of 1 in $$x$$, the $$y$$-value decreases by 4. --- Would you like further details or have any questions? ### Related Questions: 1. How do you determine the slope from a linear equation? 2. What is the relationship between the slope and the change in $$y$$-value? 3. How would the equation change if the slope were positive? 4. What does the $$y$$-intercept represent in a linear equation? 5. How can you graph a linear equation based on this table? ### Tip: When solving linear equations, always look at the slope and $$y$$-intercept to quickly understand how the graph behaves and how the values change.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Functions
Slope
Y-intercept

Formulas

y = mx + b

Theorems

Slope-intercept form of a linear equation

Suitable Grade Level

Grades 7-9

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