To determine the equation of the linear relationship given in the table, let's follow these steps:
1. Identify the Points:
- We have points (1, -1), (2, -1.5), (3, -2), and (4, -2.5).
2. Calculate the Slope (m):
- The formula for the slope [tex]\(m\)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
- Using points (1, -1) and (2, -1.5):
[tex]\[
m = \frac{-1.5 - (-1)}{2 - 1} = \frac{-1.5 + 1}{1} = \frac{-0.5}{1} = -0.5
\][/tex]
- To ensure consistency, let's verify the slope with other points. Using points (2, -1.5) and (3, -2):
[tex]\[
m = \frac{-2 - (-1.5)}{3 - 2} = \frac{-2 + 1.5}{1} = \frac{-0.5}{1} = -0.5
\][/tex]
- The slope [tex]\(m\)[/tex] remains consistent as [tex]\(-0.5\)[/tex].
3. Determine the Y-intercept (b):
- The equation of a line in slope-intercept form is [tex]\(y = mx + b\)[/tex].
- Using one of the points, say (1, -1), we can solve for [tex]\(b\)[/tex]:
[tex]\[
-1 = -0.5(1) + b
\][/tex]
[tex]\[
-1 = -0.5 + b
\][/tex]
[tex]\[
b = -1 + 0.5
\][/tex]
[tex]\[
b = -0.5
\][/tex]
4. Formulate the Slope-Intercept Equation:
- We now have the slope [tex]\(m = -0.5\)[/tex] and the y-intercept [tex]\(b = -0.5\)[/tex].
- Thus, the slope-intercept form of the equation is:
[tex]\[
y = -0.5x - 0.5
\][/tex]
5. Convert to Standard Form (Ax + By = C):
- Starting from [tex]\(y = -0.5x - 0.5\)[/tex], we need to clear the fractions and bring it to standard form.
- Multiply every term by 2 to eliminate the decimals:
[tex]\[
2y = -x - 1
\][/tex]
- Rearrange to get all terms on one side:
[tex]\[
x + 2y = -1
\][/tex]
Therefore, the equation representing the linear relationship in standard form is:
[tex]\[
x + 2y = -1
\][/tex]
