Question

Translate the phrases to a system of linear equations: y= a1*x+b1 and y=a2*x+b2 with a variables, e.g. x and y. What do the y-intercepts represent in your example? What does the solution (or intersection) represent in your example? Solve the system of equation for x and y.

Answer 1

The linear equations: y= a₁x+b₁ …(1)and y = a₂x+b₂…(2) ,with a variables x and y represent two lines. The 1^st linear equation i.e. y= a₁x+b₁ represents a line with slope a₁ and the y-intercept b₁. The 2^nd linear equation i.e. y = a₂x+b₂ represents a line with slope a₂ and the y-intercept b₂. The y-intercept is the distance from the origin, of the point where the line meets the Y-Axis.

On multiplying both the sides of the 1^st equation by a₂ and the 2^nd equation by a₁,we get a₂y= a₁a₂x+a₂ b₁…(3) and a₁y= a₁a₂x+a₁b₂…(4).

Now, on subtracting the 3^rd equation from the 4^th equation, we get a₁y- a₂y = a₁a₂x+a₁b₂-( a₁a₂x+a₂b₁) or, y(a₁-a₂) = a₁b₂- a₂b₁ so that x = (b₂-b₁)/( a₁-a₂). On substituting this value of y in the 1^st equation, we get (a₁b₂- a₂b₁)/( a₁-a₂)=a₁x+b₁ or, a₁x=(a₁b₂- a₂b₁)/(a₁-a₂)-b₁ = a₁(b₂-b₁)/(a₁-a₂) so that x = (b₂-b₁)/( a₁-a₂).

Thus, the solution is x = (b₂-b₁)/( a₁-a₂) and x = (b₂-b₁)/( a₁-a₂). These are the x and y coordinates of the point of intersection of the 2 given lines. However, it may be observed that the 2 given lines intersect, i.e. there is a solution to the 2 given linear equations only if a₁ is not equal to a₂.