Math / Algebra

Questiond a $y$ -intercept of -0.75 and a gradient of 0.75 (e) a $y$ -intercept of -2 and a gradient of 0 f a gradient of 0 and a $y$ -intercept of 4 . 5 Find the equation (in the form $a x+b y=c$ ) of a line which has: a a gradient of $-\frac{3}{2}$ and a $y$ -intercept at $(0,-0.5)$ b a $y$ -intercept of 2 and a gradient of $-\frac{3}{4}$ c a $y$ -intercept of -3 and a gradient of $\frac{4}{8}$ .

Studdy Solution

STEP 1

1. We are given various gradients (slopes) and y-intercepts for different lines.
2. We need to find the equation of each line in the form $ax + by = c$ .
3. The general form of a line equation is $y = mx + c$ , where $m$ is the gradient and $c$ is the y-intercept.

STEP 2

1. Convert the slope-intercept form to the standard form for each line.
2. Simplify the equations to match the form $ax + by = c$ .

STEP 3

For part (a), we have a gradient of $-\frac{3}{2}$ and a y-intercept at $(0, -0.5)$ .
The slope-intercept form is: $y = -\frac{3}{2}x - 0.5$
Convert to standard form: $2y = -3x - 1$ $3x + 2y = -1$

STEP 4

For part (b), we have a y-intercept of 2 and a gradient of $-\frac{3}{4}$ .
The slope-intercept form is: $y = -\frac{3}{4}x + 2$
Convert to standard form: $4y = -3x + 8$ $3x + 4y = 8$

STEP 5

For part (c), we have a y-intercept of -3 and a gradient of $\frac{4}{8}$ , which simplifies to $\frac{1}{2}$ .
The slope-intercept form is: $y = \frac{1}{2}x - 3$
Convert to standard form: $2y = x - 6$ $x - 2y = 6$

STEP 6

For part (d), we have a y-intercept of -0.75 and a gradient of 0.75.
The slope-intercept form is: $y = 0.75x - 0.75$
Convert to standard form: $4y = 3x - 3$ $3x - 4y = 3$

STEP 7

For part (e), we have a y-intercept of -2 and a gradient of 0.
The slope-intercept form is: $y = -2$
Convert to standard form: $0x + y = -2$

STEP 8

For part (f), we have a gradient of 0 and a y-intercept of 4.
The slope-intercept form is: $y = 4$
Convert to standard form: $0x + y = 4$
The equations of the lines in standard form are: (a) $3x + 2y = -1$ (b) $3x + 4y = 8$ (c) $x - 2y = 6$ (d) $3x - 4y = 3$ (e) $y = -2$ (f) $y = 4$

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Studdy Solution

STEP 1

1. We are given various gradients (slopes) and y-intercepts for different lines.
2. We need to find the equation of each line in the form $ax + by = c$ .
3. The general form of a line equation is $y = mx + c$ , where $m$ is the gradient and $c$ is the y-intercept.

STEP 2

1. Convert the slope-intercept form to the standard form for each line.
2. Simplify the equations to match the form $ax + by = c$ .

STEP 3

For part (a), we have a gradient of $-\frac{3}{2}$ and a y-intercept at $(0, -0.5)$ .
The slope-intercept form is: $y = -\frac{3}{2}x - 0.5$
Convert to standard form: $2y = -3x - 1$ $3x + 2y = -1$

STEP 4

For part (b), we have a y-intercept of 2 and a gradient of $-\frac{3}{4}$ .
The slope-intercept form is: $y = -\frac{3}{4}x + 2$
Convert to standard form: $4y = -3x + 8$ $3x + 4y = 8$

STEP 5

For part (c), we have a y-intercept of -3 and a gradient of $\frac{4}{8}$ , which simplifies to $\frac{1}{2}$ .
The slope-intercept form is: $y = \frac{1}{2}x - 3$
Convert to standard form: $2y = x - 6$ $x - 2y = 6$

STEP 6

For part (d), we have a y-intercept of -0.75 and a gradient of 0.75.
The slope-intercept form is: $y = 0.75x - 0.75$
Convert to standard form: $4y = 3x - 3$ $3x - 4y = 3$

STEP 7

For part (e), we have a y-intercept of -2 and a gradient of 0.
The slope-intercept form is: $y = -2$
Convert to standard form: $0x + y = -2$

STEP 8

For part (f), we have a gradient of 0 and a y-intercept of 4.
The slope-intercept form is: $y = 4$
Convert to standard form: $0x + y = 4$
The equations of the lines in standard form are: (a) $3x + 2y = -1$ (b) $3x + 4y = 8$ (c) $x - 2y = 6$ (d) $3x - 4y = 3$ (e) $y = -2$ (f) $y = 4$

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Studdy Solution

STEP 1

1. We are given various gradients (slopes) and y-intercepts for different lines.2. We need to find the equation of each line in the form ax+by=c ax + by = c ax+by=c.3. The general form of a line equation is y=mx+c y = mx + c y=mx+c, where m m m is the gradient and c c c is the y-intercept.

STEP 2

1. Convert the slope-intercept form to the standard form for each line.2. Simplify the equations to match the form ax+by=c ax + by = c ax+by=c.

STEP 3

For part (a), we have a gradient of −32-\frac{3}{2}−23​ and a y-intercept at (0,−0.5) (0, -0.5) (0,−0.5).The slope-intercept form is: y=−32x−0.5 y = -\frac{3}{2}x - 0.5 y=−23​x−0.5Convert to standard form: 2y=−3x−1 2y = -3x - 1 2y=−3x−1 3x+2y=−1 3x + 2y = -1 3x+2y=−1

STEP 4

For part (b), we have a y-intercept of 2 and a gradient of −34-\frac{3}{4}−43​.The slope-intercept form is: y=−34x+2 y = -\frac{3}{4}x + 2 y=−43​x+2Convert to standard form: 4y=−3x+8 4y = -3x + 8 4y=−3x+8 3x+4y=8 3x + 4y = 8 3x+4y=8

STEP 5

For part (c), we have a y-intercept of -3 and a gradient of 48\frac{4}{8}84​, which simplifies to 12\frac{1}{2}21​.The slope-intercept form is: y=12x−3 y = \frac{1}{2}x - 3 y=21​x−3Convert to standard form: 2y=x−6 2y = x - 6 2y=x−6 x−2y=6 x - 2y = 6 x−2y=6

STEP 6

For part (d), we have a y-intercept of -0.75 and a gradient of 0.75.The slope-intercept form is: y=0.75x−0.75 y = 0.75x - 0.75 y=0.75x−0.75Convert to standard form: 4y=3x−3 4y = 3x - 3 4y=3x−3 3x−4y=3 3x - 4y = 3 3x−4y=3

STEP 7

For part (e), we have a y-intercept of -2 and a gradient of 0.The slope-intercept form is: y=−2 y = -2 y=−2Convert to standard form: 0x+y=−2 0x + y = -2 0x+y=−2

STEP 8

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1. We are given various gradients (slopes) and y-intercepts for different lines.
2. We need to find the equation of each line in the form $ax + by = c$ .
3. The general form of a line equation is $y = mx + c$ , where $m$ is the gradient and $c$ is the y-intercept.

1. Convert the slope-intercept form to the standard form for each line.
2. Simplify the equations to match the form $ax + by = c$ .

For part (a), we have a gradient of $-\frac{3}{2}$ and a y-intercept at $(0, -0.5)$ .
The slope-intercept form is: $y = -\frac{3}{2}x - 0.5$
Convert to standard form: $2y = -3x - 1$ $3x + 2y = -1$

For part (b), we have a y-intercept of 2 and a gradient of $-\frac{3}{4}$ .
The slope-intercept form is: $y = -\frac{3}{4}x + 2$
Convert to standard form: $4y = -3x + 8$ $3x + 4y = 8$

For part (c), we have a y-intercept of -3 and a gradient of $\frac{4}{8}$ , which simplifies to $\frac{1}{2}$ .
The slope-intercept form is: $y = \frac{1}{2}x - 3$
Convert to standard form: $2y = x - 6$ $x - 2y = 6$

For part (d), we have a y-intercept of -0.75 and a gradient of 0.75.
The slope-intercept form is: $y = 0.75x - 0.75$
Convert to standard form: $4y = 3x - 3$ $3x - 4y = 3$

For part (e), we have a y-intercept of -2 and a gradient of 0.
The slope-intercept form is: $y = -2$
Convert to standard form: $0x + y = -2$