Question

here is a graph of x from 0 to 2 By drawing a second graph on the grid, work out an approximate solution to the simultaneous equation 3x-y=2 and 2x+y=4

Answer 1

(1.2, 1.6)

Explanation:

Rewrite both equations to make y the subject:

Equation 1: 3x - y = 2

⇒ 3x = 2 + y

⇒ y = 3x - 2

This line has a positive slope.

Equation 2: 2x + y = 4

⇒ y = -2x + 4

This line has a negative slope.

Therefore, the graph that has been plotted is y = 3x - 2

Plotting second graph

Equation of the line: y = -2x + 4

Input x = 0 into the equation and solve to find the y-intercept:

⇒ y = -2(0) + 4 = 4

Therefore, the y-intercept is (0, 4)

Input y = 0 into the equation and solve to find the x-intercept:

⇒ 0 = -2x + 4

⇒ 2x = 4

⇒ x = 2

Therefore, the x-intercept is (2, 0)

Plot the two points (0, 4) and (2, 0), and draw a straight line that passes through them.

The solution to the system is the point of intersection of the two lines. From inspection of the graph, this is (1.2, 1.6)

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Solving algebraically

Equation 1: 3x - y = 2

Equation 2: 2x + y = 4

Sum the equations to eliminate y:

`\begin{aligned}& 3x - y & = 2\\ +\quad& 2x + y & = 4\\ \cline{1-3}& 5x & = 6\end{aligned}`

Solve for x:

`\implies x=\frac65=1.2`

Substitute found value of x into one of the equations and solve for y:

`\implies y = 3\left(1.2) - 2=1.6`

So the solution (point of intersection) of the system of equations is:
(1.2, 1.6)