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The line
M of equation
2x-3y=5 crosses the
x-axis at point
A.
(a) Find the coordinates of point
A.
(b) Given that the line
M passes through the point
(b,-2.5), find the value of
b.
(c) State the gradient of the line
M.

$5$ . The line $M$ of equation $2 x-3 y=5$ crosses the $x$ -axis at point $A$ . $\newline$ (a) Find the coordinates of point $A$ . $\newline$ (b) Given that the line $M$ passes through the point $(b,-2.5)$ , find the value of $b$ . $\newline$ (c) State the gradient of the line $M$ .

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Write the equation of a linear function

Full solution

Q.

5

. The line

M

of equation

2 x-3 y=5

crosses the

x

-axis at point

A

.

\newline

(a) Find the coordinates of point

A

.

\newline

(b) Given that the line

M

passes through the point

(b,-2.5)

, find the value of

b

.

\newline

(c) State the gradient of the line

M

.

Find x-intercept: To find where the line crosses the x-axis, set $y=0$ in the equation $2x-3y=5$ and solve for $x$ . $\newline$ $2x - 3(0) = 5$ $\newline$ $2x = 5$ $\newline$ $x = \frac{5}{2}$ $\newline$ So, point A is $(\frac{5}{2}, 0)$ .
Find value of b: Now, to find the value of $b$ when the line passes through $(b, -2.5)$ , substitute $y=-2.5$ into the equation and solve for $x$ . $2x - 3(-2.5) = 5$ $2x + 7.5 = 5$ $2x = 5 - 7.5$ $2x = -2.5$ $x = -2.5/2$ $x = -1.25$ So, $(b, -2.5)$ $0$ .
Calculate gradient: To find the gradient of the line, rearrange the equation into slope-intercept form $y = mx + b$ . $2x - 3y = 5$ $-3y = -2x + 5$ $y = \frac{-2x + 5}{-3}$ $y = \frac{2}{3}x - \frac{5}{3}$ The gradient $m$ is $\frac{2}{3}$ .

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