Graph the line. $$ y=-5 x+1 $$

Certainly! Let's graph the linear equation: \[ y = -5x + 1 \] ### Step 1: Identify the Components of the Equation The equation is in the slope-intercept form: \[ y = mx + b \] Where: - $$ m $$ is the **slope** of the line. - $$ b $$ is the **y-intercept** (the point where the line crosses the y-axis). For your equation: - **Slope ($$ m $$)**: $$-5$$ - **Y-intercept ($$ b $$)**: $$1$$ ### Step 2: Plot the Y-Intercept The y-intercept is the point where the line crosses the y-axis ($$ x = 0 $$). \[ (0, b) = (0, 1) \] Plot the point **(0, 1)** on the coordinate plane. ### Step 3: Use the Slope to Find Another Point The slope $$ m = -5 $$ can be interpreted as a ratio $$\frac{\Delta y}{\Delta x}$$. Specifically, $$-5$$ means that for every **1 unit** you move **right** along the x-axis ($$ \Delta x = 1 $$), you move **down** **5 units** along the y-axis ($$ \Delta y = -5 $$). Starting from the y-intercept (0, 1): - Move **1 unit right** to $$ x = 1 $$. - Move **5 units down** to $$ y = 1 - 5 = -4 $$. Plot the point **(1, -4)**. ### Step 4: Draw the Line Using a ruler or straight edge: 1. Draw a straight line that passes through both points **(0, 1)** and **(1, -4)**. 2. Extend the line in both directions. 3. Optionally, add arrows on both ends to indicate that the line continues indefinitely. ### Step 5: (Optional) Find Additional Points for Accuracy For greater precision, you can plot more points by choosing different x-values: - **When $$ x = -1 $$:** \[ y = -5(-1) + 1 = 5 + 1 = 6 \quad \Rightarrow \quad (-1, 6) \] - **When $$ x = 2 $$:** \[ y = -5(2) + 1 = -10 + 1 = -9 \quad \Rightarrow \quad (2, -9) \] Plot these additional points and ensure they all lie on the same straight line. ### Visual Representation While I can't provide a visual graph directly, here's a simple ASCII representation: ``` y | 6 | * 5 | 4 | 3 | 2 | 1 | * 0 |---------*--------- x | (0,1) (1,-4) ``` *Note:* For a more accurate and detailed graph, you can use graphing tools like [Desmos](https://www.desmos.com/calculator), [GeoGebra](https://www.geogebra.org/), or graphing calculators. ### Summary - **Y-Intercept:** (0, 1) - **Using Slope ($$ m = -5 $$):** From (0, 1), go right 1 and down 5 to reach (1, -4). - **Draw the line** through these points, extending in both directions. This line will slope downward steeply from left to right due to the negative slope of $$-5$$. To graph the line $$ y = -5x + 1 $$, plot the y-intercept at (0, 1). Then, use the slope of -5 to find another point by moving 1 unit right and 5 units down to (1, -4). Draw a straight line through these points, extending in both directions.

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Pregunta

Graph the line.

Ask by Lyons Marsh. in the United States

Jan 24,2025

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To graph the line

, plot the y-intercept at (0, 1). Then, use the slope of -5 to find another point by moving 1 unit right and 5 units down to (1, -4). Draw a straight line through these points, extending in both directions.

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To graph the line

, start by identifying two key points. First, when

; this gives you the y-intercept at (0, 1). Next, choose

, which yields

; now you have another point at (1, -4). Plot these two points and draw a straight line through them, which will have a steep negative slope reflecting the ‘-5’ in the equation.

Now that you’ve drawn your line, it’s time to consider how you would interpret this in real life! The equation represents how one variable (y) decreases rapidly as another variable (x) increases. This could model various scenarios, such as the relationship between the number of hours worked and the amount of money left to spend, where more hours result in significantly less leisure time—so keep an eye on those slopes both on graphs and in life!

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