Question

Solve the following simultaneous equation graphically.
5x – 6y + 30 = 0 ; 5x + 4y – 20 = 0

Answer 1

Step 1: Rewrite the Equations in Slope-Intercept Form

First, we need to convert each equation to the slope-intercept form (y = mx + b) to facilitate graphing.

Equation 1: 5x − 6y + 30 = 0

Solve for y:

5x − 6y + 30 = 0

−6y = −5x − 30

y = `5/6​x` + 5

Equation 2: 5x + 4y − 20 = 0

Solve for y:

5x + 4y − 20 = 0

4y = `-5/4x` + 20

y = `-5/4x` + 5

Now we have the equations in the slope-intercept form:

y = `5/6x` + 5

Step 2: Plot the Equations on a Graph

To plot these lines, we need to find the points where they intersect the x-axis and y-axis.

Equation 1: y = `5/6x` + 5

When x = 0:

y = `5/6` ​(0) + 5 = 5

So, the y-intercept is (0,5).

When y = 0:

0 = `5/6x` + 5

`5/6x` = −5

x = −6

So, the x-intercept is (−6,0).

Equation 2: y = −`5/4x` + 5

When x = 0:

y = −`5/4`​(0) + 5 = 5

So, the y-intercept is (0,5).

When y = 0:

0 = −`5/4x`​​ + 5

− `5/4x` = −5

x = 4

So, the x-intercept is (4,0).

Step 3: Draw the Lines on the Graph

Line 1: y = `5/6x` + 5

Passes through (0,5) and (−6,0)

Line 2: y=−`5/4x`+5

Passes through (0,5) and (4,0)

Step 4: Find the Intersection Point

Plot both lines on the same graph. The intersection point of the lines is the solution to the simultaneous equations.

The lines intersect at (0,5).

Step 5: Verify the Solution

Verify the intersection point by substituting (0,5) into the original equations.

For 5x − 6y + 30 = 0:

5(0) − 6(5) + 30 = 0

−30 + 30 = 0

0 = 0

For 5x + 4y − 20 = 0:

5(0) + 4(5) − 20 = 0

20 − 20 = 0

0 = 0

Both equations are satisfied by the point (0,5).

The solution to the simultaneous equations 5x − 6y + 30 = 0 and 5x + 4y − 20 = 0 is: (0,5)