Find the intersection of the lines y = x +1 and 3y + 2x = 13 without drawing the graph.

The intersection point is (2, 3)

Step-by-step explanation:

The given (linear) equations for the point of intersection is required are;

y = x + 1...(1)

3·y + 2·x = 13...(2)

To find the point of intersection without using graph, we note that the point of intersection is the point at which the values of both equations are equal

We find the values of 'x' and y-coordinates at the point of intersection as follows;

Making 'y' the subject of equation (2) gives;

y = (13 - 2·x)/3

When both lines intersect, the 'x' and y-values are the same;

y = x + 1 for the first line, and y = (13 - 2·x)/3 for the second line, therefore, at the intersection point, we have;

y = y

x + 1 = (13 - 2·x)/3

3·(x + 1) = 13 - 2·x

3·x + 3 = 13 - 2·x

3·x + 2·x = 13 - 3

3·x + 2·x = 5·x = 13 - 3 = 10

∴ 5·x = 10

Therefore, at the point of intersection, x = 10/5 = 2

x = 2

'y' is given in equation (1) by y = x + 1, therefore, at the point of intersection, 'x = 2', we get;

y = x + 1

∴ The value of 'y' at the point of intersection is y = 2 + 1 = 3

y = 3

Therefore, the (x, y) coordinate at the point of intersection is (2, 3) which can be written as follows;

The point of intersection = (2, 3).

m∠cdf = 43° , m∠gdf = 114°, m∠gde = 23°, m∠edh = 23° , m∠cdh = 157°

step-by-step explanation:

since, it is given that de bisects the angle gdh ⇒ de divides the angle gdh into two equal angles.

⇒ m∠gde = m∠edh

⇒ 8·x   - 1 = 6·x + 5

⇒ x = 3

so, m∠gde = 8·x - 1 = 23°

and m∠edh = 23°

now, it is given ∠cde is an straight angle so, m∠cde = 180°

⇒ m∠cdf + m∠fdg + m∠gde = 180°

⇒ 43° + m∠fdg + 23° = 180°

⇒ m∠fdg = 114°

again, m∠cde = m∠cdh + m∠edh = 180°

m∠cdh + 23° = 180°

⇒ m∠cdh = 157°