What is the standard form of the following equation?

Question

What is the standard form of the following equation?
y = −2/5x + 4
Use integers for A, B, and C. Enter your answer in the box.

GPT-4o mini · Answer

To convert the equation $ y = -\frac{2}{5}x + 4 $ into standard form $ Ax + By = C $, we need to rearrange the equation so that all terms involving $ x $ and $ y $ are on one side and the constant is on the other side. Starting with the original equation: $$ y = -\frac{2}{5}x + 4 $$ We can eliminate the fraction by multiplying every term by 5: $$ 5y = -2x + 20 $$ Next, we'll rearrange it to standard form by adding $ 2x $ to both sides: $$ 2x + 5y = 20 $$ Now, the equation is in the standard form $ Ax + By = C $, where $ A = 2 $, $ B = 5 $, and $ C = 20 $. Thus, the standard form of the equation is: $$ \boxed{2x + 5y = 20} $$

Bornana · Answer

What is the slope-intercept form of the following equation? 10x + 4y = 16 Enter your answer in the box.

GPT-4o mini · Answer

To convert the equation $ 10x + 4y = 16 $ into slope-intercept form, which is given by $ y = mx + b $ (where $ m $ is the slope and $ b $ is the y-intercept), we need to solve for $ y $. Starting with the original equation: $$ 10x + 4y = 16 $$ First, we isolate the term containing $ y $ by subtracting $ 10x $ from both sides: $$ 4y = -10x + 16 $$ Next, we solve for $ y $ by dividing every term by 4: $$ y = -\frac{10}{4}x + \frac{16}{4} $$ This simplifies to: $$ y = -\frac{5}{2}x + 4 $$ Therefore, the slope-intercept form of the equation is: $$ \boxed{y = -\frac{5}{2}x + 4} $$

Bornana · Answer

Use the Line tool to graph the equation. 3x − 2y = 12

GPT-4o mini · Answer

While I cannot directly create or manipulate graphs, I can help you understand how to graph the equation $3x - 2y = 12$ using information you can plot on graph paper or a graphing tool.

Steps to Graph the Equation:

Convert to Slope-Intercept Form: To find the slope and y-intercept, we can convert the equation to the slope-intercept form $y = mx + b$.

Start with: \[ 3x - 2y = 12 \]

Rearranging to solve for $y$: \[ -2y = -3x + 12 \] \[ 2y = 3x - 12 \] \[ y = \frac{3}{2}x - 6 \]

Slope (m): $\frac{3}{2}$
Y-intercept (b): -6 (point where the line crosses the y-axis)
Plot the Y-Intercept: Start by plotting the y-intercept:
- Point: (0, -6)
Use the Slope to Find Another Point: From the y-intercept, use the slope to find another point. The slope $\frac{3}{2}$ means that for every 2 units you move to the right (positive x-direction), you'll move up 3 units (positive y-direction).
- From (0, -6), if you move 2 units right to (2, -6) and then 3 units up, you reach the point (2, -3).
Plot the Second Point:
- Point: (2, -3)
Draw the Line: With the points plotted, draw a straight line through (0, -6) and (2, -3). Extend the line in both directions and add arrows on both ends.

Additional Points for Accuracy:

To ensure your line is accurate, you may want to calculate and plot more points by substituting values for $x$ and solving for $y$.

If $x = 0$, then $y = -6$ → Point (0, -6)
If $x = 2$, then $y = -3$ → Point (2, -3)
If $x = 4$: \[ y = \frac{3}{2}(4) - 6 = 6 - 6 = 0 \quad \text{→ Point (4, 0)} \]
If $x = 6$: \[ y = \frac{3}{2}(6) - 6 = 9 - 6 = 3 \quad \text{→ Point (6, 3)} \]

By plotting additional points, you can ensure that your graph of the equation is accurate.

Bornana · Answer

Graph y and x. y=14x+2

GPT-4o mini · Answer

To graph the equation $y = 14x + 2$, follow these steps: ### Step 1: Identify the Slope and Y-Intercept The equation is already in slope-intercept form $y = mx + b$, where: - $m$ (slope) = 14 - $b$ (y-intercept) = 2 This means the line crosses the y-axis at $ (0, 2) $. ### Step 2: Plot the Y-Intercept - Start by plotting the point $ (0, 2) $ on the graph. This is where the line crosses the y-axis. ### Step 3: Use the Slope to Find Another Point The slope of the line is 14, which can be interpreted as: - For every 1 unit you move to the right (increase in $x$), you move up 14 units (increase in $y$). Starting from the y-intercept $ (0, 2) $: - Move 1 unit to the right to $ (1, 2) $. - Move up 14 units to reach $ (1, 16) $. ### Step 4: Plot the Second Point - Plot the point $ (1, 16) $. ### Step 5: Draw the Line With the two points $ (0, 2) $ and $ (1, 16) $ plotted: - Draw a straight line through these points and extend it in both directions with arrows on each end. ### Additional Points for Accuracy You can calculate additional points by plugging in values for $x$ and solving for $y$: - If $x = -1$: $$ y = 14(-1) + 2 = -14 + 2 = -12 \quad \text{→ Point} (-1, -12) $$ - If $x = 2$: $$ y = 14(2) + 2 = 28 + 2 = 30 \quad \text{→ Point} (2, 30) $$ ### Final Note With these points, you can accurately graph $y = 14x + 2$. If using graphing software or graph paper, make sure to label the axes appropriately and extend the line beyond the plotted points. The line will rise steeply due to the high slope of 14.