Questions: Write an equation of the line that passes through (-1,4) and is parallel to the line defined by 3x+y=-3. Write the answer in slope-intercept form (if possible) and in standard form (Ax+By=C) with smallest integer coefficients. Use the "Cannot be written" button, if applicable.

Write an equation of the line that passes through (-1,4) and is parallel to the line defined by 3x+y=-3. Write the answer in slope-intercept form (if possible) and in standard form (Ax+By=C) with smallest integer coefficients. Use the "Cannot be written" button, if applicable.

Transcript text: Write an equation of the line that passes through $(-1,4)$ and is parallel to the line defined by $3 x+y=-3$. Write the answer in slope-intercept form (if possible) and in standard form $(A x+B y=C)$ with smallest integer coefficients. Use the "Cannot be written" button, if applicable.

Solution

Solution Steps

Step 1: Determine the Slope of the Given Line

The equation of the given line is $3x + y = -3$. To find the slope, we can rewrite this equation in slope-intercept form $y = mx + b$: \[ y = -3x - 3 \] From this, we see that the slope $m$ of the given line is $-3$.

Step 2: Use the Same Slope for the New Line

Since we are looking for a line that is parallel to the given line, the slope of the new line will also be $-3$.

Step 3: Use Point-Slope Form to Find the New Line

We will use the point-slope form of the equation of a line, which is given by: \[ y - y_1 = m(x - x_1) \] Substituting the point $(-1, 4)$ and the slope $-3$: \[ y - 4 = -3(x + 1) \]

Step 4: Convert to Slope-Intercept Form

Rearranging the equation from Step 3 to isolate $y$: \[ y - 4 = -3x - 3 \implies y = -3x + 1 \]

Step 5: Convert to Standard Form

To express the equation in standard form $Ax + By = C$, we rearrange the slope-intercept form: \[ 3x + y = 1 \] This is the standard form of the equation with the smallest integer coefficients.