Question

Write an equation of the line that passes through (3,-4) and is parallel to the line defined by 4x=5y+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.

Answer 1

To find the equation of a line parallel to 4x=5y+3 that passes through (3,-4), we first rewrite the given equation in slope-intercept form to determine the slope is 4/5. The new line has the same slope, represented in slope-intercept form as y=(4/5)x-(32/5). Converted to standard form, our equation is 4x-5y=32.

Step-by-step explanation:

To find an equation of a line parallel to 4x=5y+3, we must first put this equation in slope-intercept form, which is y=mx+b where m represents slope and b the y-intercept. Dividing by 5, the slope-intercept form would be y=(4/5)x-3/5, which means that the slope of this line is 4/5.

Since parallel lines have the same slope, the line we're looking for will also have a slope of 4/5. Using the point-slope form of a line (y-y1)=m(x-x1), with the point (3,-4), we will have:

y+4=(4/5)(x-3)

Distributing the (4/5) we get y+4=(4/5)x-(12/5). Solving for y leads to the slope-intercept form of our line: y=(4/5)x-(32/5).

To convert this into standard form, Ax+By=C, we need to eliminate any fractions and make A,B, and C the smallest integers possible. Multiplying both sides of the equation by 5 to clear the fraction gives us:

5y=4x-32

Subtracting 4x from both sides yields the standard form:

-4x+5y=-32

Ensuring A is positive, as is the convention, we multiply by -1 to get:

4x-5y=32

Answer 2

The equation of the line parallel to 4x=5y+3 passing through (3,-4) is y=(4/5)x - 32/5 in slope-intercept form and 4x-5y=32 in standard form.

Step-by-step explanation:

To find the equation of a line parallel to another line, we need to find the slope of the given line.

The given line 4x=5y+3 can be rewritten in slope-intercept form as y=(4/5)x-3/5, where the slope is 4/5.

Since the parallel line has the same slope, the equation will be y=(4/5)x + b. To find the value of b, we can substitute the coordinates (3,-4) into the equation and solve for b.

Substituting the coordinates into the equation, -4=(4/5)(3)+b.

Simplifying, -4=12/5+b. Subtracting 12/5 from both sides, -4-12/5=b.

Common denominator is 5, so -20/5-12/5=b, which gives b=-32/5.

Therefore, the equation of the line that passes through (3,-4) and is parallel to the line 4x=5y+3 is y=(4/5)x - 32/5 in slope-intercept form and 4x-5y=32 in standard form with smallest integer coefficients.