Question

score: 15/20 answered: 15/20 question 16 give the slope and the y intercept of the line -10x - 6y = -5. slope = y intercept = (0, ) enter your answers as integers or as reduced fractions in the form a/b. question help: video submit question

Explanation:

Step1: Rewrite in slope-intercept form

We start with the equation \(-10x - 6y=-5\). We want to solve for \(y\) to get it into the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\)-intercept. First, we add \(10x\) to both sides:
\(-6y = 10x - 5\)

Step2: Solve for \(y\)

Now, we divide every term by \(-6\) to isolate \(y\):
\(y=\frac{10x}{-6}-\frac{-5}{-6}\)
Simplify the fractions:
\(y = -\frac{5}{3}x-\frac{5}{6}\) Wait, no, let's check that division again. Wait, \(-5\) divided by \(-6\) is \(\frac{5}{6}\)? Wait, no, original equation: \(-10x - 6y=-5\). Let's do it again.

Starting over: \(-10x - 6y=-5\)

Add \(10x\) to both sides: \(-6y = 10x - 5\)

Now divide both sides by \(-6\):

\(y=\frac{10x}{-6}+\frac{5}{6}\) (because \(-5\) divided by \(-6\) is \(\frac{5}{6}\))

Simplify \(\frac{10}{-6}\) to \(-\frac{5}{3}\). So:

\(y = -\frac{5}{3}x+\frac{5}{6}\)

Wait, that's correct. So the slope \(m\) is \(-\frac{5}{3}\) and the \(y\)-intercept \(b\) is \(\frac{5}{6}\). Wait, let's verify:

Original equation: \(-10x -6y = -5\)

Let's plug in \(x = 0\) to find the \(y\)-intercept. When \(x = 0\), we have \(-6y=-5\), so \(y=\frac{5}{6}\), which matches. And the slope is the coefficient of \(x\) when in \(y = mx + b\) form, which is \(-\frac{5}{3}\).

Wait, let's do the algebra again carefully:

Starting with \(-10x - 6y=-5\)

Add \(10x\) to both sides: \(-6y = 10x - 5\)

Divide both sides by \(-6\):

\(y=\frac{10x}{-6}+\frac{5}{6}\) (because \(-5\) divided by \(-6\) is \(\frac{5}{6}\))

Simplify \(\frac{10}{-6}\) to \(-\frac{5}{3}\), so \(y = -\frac{5}{3}x+\frac{5}{6}\)

Yes, that's correct. So the slope is \(-\frac{5}{3}\) and the \(y\)-intercept is \(\frac{5}{6}\). Wait, but let's check the sign again. When we have \(-6y = 10x - 5\), dividing by \(-6\) gives \(y=\frac{10x}{-6}-\frac{-5}{-6}\)? Wait, no, that's the mistake. Wait, \(-6y = 10x - 5\)

So \(y=\frac{10x}{-6}+\frac{5}{6}\) because \(-5\) divided by \(-6\) is \(\frac{5}{6}\). Yes, because \(-6y = 10x -5\) => \(y = \frac{10x}{-6}+\frac{5}{6}\) (since \(-5/-6 = 5/6\)). So that's correct. So slope is \(-\frac{5}{3}\) and \(y\)-intercept is \(\frac{5}{6}\). Wait, but let's check with the original equation. Let's plug \(x = 0\) into the original equation: \(-10(0)-6y=-5\) => \(-6y=-5\) => \(y = 5/6\), which matches. And the slope: let's take two points. Let's find when \(x = 0\), \(y = 5/6\). When \(x = 3\), \(y = -\frac{5}{3}(3)+\frac{5}{6}=-5+\frac{5}{6}=-\frac{25}{6}\). Now, let's plug \(x = 3\) into the original equation: \(-10(3)-6y=-5\) => \(-30 -6y=-5\) => \(-6y=25\) => \(y = -25/6\), which matches. So the slope between \((0, 5/6)\) and \((3, -25/6)\) is \(\frac{-25/6 - 5/6}{3 - 0}=\frac{-30/6}{3}=\frac{-5}{3}\), which is the slope we found. So that's correct.

Wait, but earlier I had a sign error when I first did the division, but corrected it. So the slope is \(-\frac{5}{3}\) and the \(y\)-intercept is \(\frac{5}{6}\). Wait, but let's check the equation again. Wait, original equation: \(-10x -6y=-5\). Let's rearrange to \(6y = -10x +5\) (multiplying both sides by -1: \(10x +6y =5\), then \(6y = -10x +5\), then \(y = -\frac{10}{6}x+\frac{5}{6}=-\frac{5}{3}x+\frac{5}{6}\). Yes, that's correct. So slope is \(-\frac{5}{3}\), \(y\)-intercept is \(\frac{5}{6}\).

Wait, but the problem says "Enter your answers as integers or as reduced fractions in the form A/B." So slope is \(-\frac{5}{3}\) and \(y\)-intercept is \(\frac{5}{6}\).

Wait, but let's do the algebra again step by step:

1. Start with \(-10x - 6y = -5\)
2. Add \(10x\) to both sides: \(-6y = 10x - 5\)
3. Divide both sides by \(-6\): \(y = \frac{10x}{-6} + \frac{-5}{-6}\)
4. Simplify: \(y = -\frac{5}{3}x + \frac{5}{6}\)

Yes, that's correct. So slope \(m = -\frac{5}{3}\), \(y\)-intercept \(b = \frac{5}{6}\).

Answer:

Slope = \(-\frac{5}{3}\)

\(y\) intercept = \((0, \frac{5}{6})\)