Question

equations of linear relations lesson #3: the general form equation ax + by + c = 0
match each equation on the left with the correct characteristic of the graph of the equation on the right. each characteristic may be used once, more than once, or not at all.
equation
i) 6x - 2y + 5 = 0
ii) 2x - 5y = 0
iii) x + 3y + 6 = 0
iv) x - 4y + 10 = 0
v) 2x - y - 5 = 0
characteristic
a. slope = -1/3
b. y-intercept = -5/2
c. passes through (-10, -4)
d. slope = 0
e. y-intercept = 5/2
f. perpendicular to y = 5/2 x - 3
g. x-intercept = 5/2

Explanation:

For Equation (i): \( 6x - 2y + 5 = 0 \)

Step 1: Convert to slope - intercept form (\(y = mx + b\))

We start with the equation \(6x - 2y+5 = 0\). First, we isolate the \(y\) - term.
Subtract \(6x\) and \(5\) from both sides: \(-2y=-6x - 5\)
Then divide both sides by \(-2\): \(y = 3x+\frac{5}{2}\)
The slope \(m = 3\) and the \(y\) - intercept \(b=\frac{5}{2}\)

Step 2: Check the \(y\) - intercept

From the slope - intercept form \(y = 3x+\frac{5}{2}\), the \(y\) - intercept is \(\frac{5}{2}\), which matches characteristic E.

Step 3: Check if it passes through \((- 10,-4)\)

Substitute \(x=-10\) and \(y = - 4\) into the original equation \(6x-2y + 5\):
\(6\times(-10)-2\times(-4)+5=-60 + 8+5=-47
eq0\), so it does not pass through \((-10,-4)\)

Step 4: Check the slope

The slope is \(3\), not related to A (\(m =-\frac{1}{3}\)) or D (\(m = 0\))

Step 5: Check the \(x\) - intercept

To find the \(x\) - intercept, set \(y = 0\) in \(6x-2y + 5 = 0\)
\(6x+5 = 0\Rightarrow x=-\frac{5}{6}
eq\frac{5}{2}\)

Step 6: Check perpendicularity

The slope of the line \(y=\frac{5}{2}x - 3\) is \(\frac{5}{2}\). The slope of a line perpendicular to it should be \(-\frac{2}{5}\) (since the product of slopes of perpendicular lines is \(- 1\), \(m_1\times m_2=-1\), if \(m_1=\frac{5}{2}\), then \(m_2 =-\frac{2}{5}\)), and our slope is \(3\), so not perpendicular.

For Equation (ii): \(2x - 5y=0\)

Step 1: Convert to slope - intercept form

Start with \(2x-5y = 0\). Isolate \(y\):
\(-5y=-2x\), then \(y=\frac{2}{5}x\)
The slope \(m=\frac{2}{5}\) and \(y\) - intercept \(b = 0\) (passes through the origin)

Step 2: Check if it passes through \((-10,-4)\)

Substitute \(x=-10\) and \(y=-4\) into \(2x - 5y\):
\(2\times(-10)-5\times(-4)=-20 + 20=0\), so the line \(2x - 5y = 0\) passes through \((-10,-4)\), which matches characteristic C.

For Equation (iii): \(x + 3y+6 = 0\)

Step 1: Convert to slope - intercept form

Start with \(x + 3y+6 = 0\). Isolate \(y\):
\(3y=-x - 6\), then \(y=-\frac{1}{3}x-2\)
The slope \(m =-\frac{1}{3}\), which matches characteristic A.

For Equation (iv): \(x - 4y+10 = 0\)

Answer:

i) - E
ii) - C
iii) - A
iv) - (No match from the given options for the main characteristics we checked, but if we re - check, maybe other properties. But based on the above)
v) - G

(Note: For a more comprehensive match, we can also check other properties. For example, for equation (iv), let's re - check the \(y\) - intercept: from \(y=\frac{1}{4}x+\frac{5}{2}\), \(y\) - intercept is \(\frac{5}{2}\), but E is already used by (i). For equation (iv), if we check the point \((-10,-4)\), it does not satisfy. If we check the \(x\) - intercept, \(x=-10\) when \(y = 0\), which is not \(\frac{5}{2}\).)