Question1) Determine if the ordered pair $(3,4)$ is a solution to the equation $6 x-4 y=6$ .
Find a point that does belong to this line. 2) Find the equation of the line that passes through the point $(6,4)$ and has a slope of 2.5. 3) Find the slopes and $y$ -intercepts for each of the following lines: (a) $2 x+5 y=23$ (b) $2 x-3 y-8=0$ 4) Find the slopes and $y$ -intercepts for each of the line that passes through the following two points: $P(0.6,-1.3) \text { and } Q(2.3,-2.4)$

Studdy Solution

STEP 1

1. We need to verify if a given point satisfies a linear equation.
2. We need to find a point on a given line.
3. We need to find the equation of a line given a point and a slope.
4. We need to find the slope and $y$ -intercept of given linear equations.
5. We need to find the slope and $y$ -intercept of a line passing through two points.

STEP 2

1. Verify if the point $(3,4)$ is a solution to the equation $6x - 4y = 6$ .
2. Find a point that belongs to the line $6x - 4y = 6$ .
3. Find the equation of a line given a point and a slope.
4. Determine the slope and $y$ -intercept of given linear equations.
5. Determine the slope and $y$ -intercept of a line passing through two points.

STEP 3

Substitute $x = 3$ and $y = 4$ into the equation $6x - 4y = 6$ .
$6(3) - 4(4) = 18 - 16 = 2$
Since $2 \neq 6$ , the point $(3,4)$ is not a solution to the equation.

STEP 4

To find a point on the line $6x - 4y = 6$ , choose a value for $x$ and solve for $y$ .
Let $x = 0$ :
$6(0) - 4y = 6$ $-4y = 6$ $y = -\frac{3}{2}$
The point $(0, -\frac{3}{2})$ is on the line.

STEP 5

The equation of a line with slope $m = 2.5$ passing through $(6,4)$ is given by the point-slope form:
$y - y_1 = m(x - x_1)$
Substitute $m = 2.5$ , $x_1 = 6$ , and $y_1 = 4$ :
$y - 4 = 2.5(x - 6)$ $y - 4 = 2.5x - 15$ $y = 2.5x - 11$

STEP 6

(a) Rearrange $2x + 5y = 23$ to slope-intercept form $y = mx + b$ :
$5y = -2x + 23$ $y = -\frac{2}{5}x + \frac{23}{5}$
Slope $m = -\frac{2}{5}$ , $y$ -intercept $b = \frac{23}{5}$ .
(b) Rearrange $2x - 3y - 8 = 0$ to slope-intercept form:
$-3y = -2x + 8$ $y = \frac{2}{3}x - \frac{8}{3}$
Slope $m = \frac{2}{3}$ , $y$ -intercept $b = -\frac{8}{3}$ .

STEP 7

To find the slope of the line through points $P(0.6, -1.3)$ and $Q(2.3, -2.4)$ :
$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2.4 + 1.3}{2.3 - 0.6} = \frac{-1.1}{1.7} \approx -0.647$
Using point-slope form with point $P(0.6, -1.3)$ :
$y + 1.3 = -0.647(x - 0.6)$ $y = -0.647x + 0.3882 - 1.3$ $y = -0.647x - 0.9118$
Slope $m \approx -0.647$ , $y$ -intercept $b \approx -0.9118$ .

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