Find the coordinates of the 𝑦-intercept and the slope of the straight line whose equation is negative five over four 𝑥 plus two 𝑦 equals nine.
To answer this question, we can recall that the slope and the 𝑦-intercept of a line can be found really easily if the line is in slope-intercept form, which is 𝑦 equals 𝑚𝑥 plus 𝑐. If the equation of a straight line is in this form, then the value of 𝑚 gives the slope of the line and the value of 𝑐 gives the 𝑦-coordinate of the 𝑦-intercept of the line.
So to answer this question, we need to take the equation of the line in the format that has been given and rearrange to make 𝑦 the subject. First, we notice that 𝑥 and 𝑦 are both on the same side of the equation. So our first step is going to be to add five over four 𝑥 to both sides. This will cancel out the negative five over four 𝑥 on the left of the equation, and on the right of the equation, we’ll now have a positive five over four 𝑥. So the equation becomes two 𝑦 is equal to five over four 𝑥 plus nine.
The equation isn’t quite in the required form yet as we have two 𝑦 where we want to have just 𝑦 or one 𝑦. So we need to divide both sides of the equation by two. So two 𝑦 divided by two gives 𝑦 and nine divided by two we can just write as nine over two.
But let’s think about how to deal with that fractional coefficient of 𝑥. We have five over four divided by two, which we can think of as five over four divided by two over one. To divide by a fraction, remember we flip or invert the fraction and we change the divide to a multiply. So we have five over four multiplied by one over two.
To multiply fractions, we multiply the numerators together and multiply the denominators together. So this gives five over eight. The rearranged equation of our line then is 𝑦 is equal to five over eight 𝑥 plus nine over two. And now, we’re in a position to find the slope and the coordinates of the 𝑦-intercept of this line.
The slope of the line is the coefficient or the value in front of the 𝑥. So the slope is five over eight. The 𝑦-coordinate of the 𝑦-intercept is nine over two. And remember this is a point on the 𝑦-axis. So it has an 𝑥-coordinate of zero.
The coordinates of the 𝑦-intercept are zero, nine over two and the slope of the line is five over eight.