Question

how many solutions does the following system have and if any what is the solution. If there are solution, find the solution.

Answer 1

$t(x)=-\frac{2}{3}x+1$$s(x)=\frac{4}{5}x+\frac{4}{15}$

When you have a swystem of lineal equation, you can find the number of solutios knowing the value of the slope (m) in each equation:

If both slopes are the same there are no solution. (m1=m2)

If the slopes are different there are one solution. (m1≠m2)

If the eqations are the same there are infinite solutions. (f(x)=g(x))

You can identify the slope (m) in a lineal equation in slope-intercept form (y=mx+b) as the coeffient of the x (the number on the left of the x).

The slope in t(x) is -2/3

The slope in s(x) is 4/5

As the slopes are different adn the equation also are different there are one solution.

To find the solution of a system of lineal equations you equal the equations:

In this case:

$t(x)=s(x)$$-\frac{2}{3}x+1=\frac{4}{5}x+\frac{4}{15}$

You clear the x and the value you get is the solution:

$-\frac{2}{3}x+1-1=\frac{4}{5}x+\frac{4}{15}-1$$-\frac{2}{3}x=\frac{4}{5}x+(\frac{4-15}{15})$$-\frac{2}{3}x=\frac{4}{5}x-\frac{11}{15}$$-\frac{2}{3}x-\frac{4}{5}x=\frac{4}{5}x-\frac{4}{5}x-\frac{11}{15}$$(\frac{-10-12}{15})x=-\frac{11}{15}$$-\frac{22}{15}x=-\frac{11}{15}$$(-\frac{15}{22})(-\frac{22}{15}x)=(-\frac{15}{22})(-\frac{11}{15})$$x=\frac{11}{22}=0.5$

You can use the value of x to find the y coordinate of the solution.

$y=-\frac{2}{3}(0.5)+1$$y=-\frac{1}{3}+1=\frac{-1+3}{3}=\frac{2}{3}=0.66$

then, the solution is x=0.5 and the coordinates of the solution are:

(0.5 , 0.66)