Question

How to do this solving linear systems by multiplying first?

Answer 1

The first thing we have to do is formulate the equations for this we identify televisions as tv and cable boxes as cb

`\begin{gathered} 1\cdot tv+1\cdot cb=1\to(1) \\ 2\cdot tv+10\cdot cb=4\to(2) \end{gathered}`

Now to solve the system of equations, one of the methods is to isolate one of the unknowns in one of the equations and replace it in the other equation.

Then we solve tv from equation (1) and replace it in (2)

`\begin{gathered} tv=1-cb\to(1) \\ 2\cdot(1-cb)+10\cdot cb=4\to(1)into(2)_{} \\ 2-2cb+10cb=4 \\ 10cb-2cb=4-2 \\ 8cb=2 \\ cb=(2)/(8) \\ cb=(1)/(4) \end{gathered}`

We can also solve the system of equations using addition and subtraction between them in the following way

`\begin{gathered} 1\cdot tv+1\cdot cb=1\to(1) \\ 2\cdot tv+10\cdot cb=4\to(2) \end{gathered}`

We multiply (1) by 2

`\begin{gathered} 2\cdot(1\cdot tv+1\cdot cb)=2\to(1) \\ 2tv+2cb=2\to2\cdot(1) \end{gathered}`

Now we subtract the equation (1) multiplied by 2 from equation (2)

`\begin{gathered} 2\cdot tv+10\cdot cb=4\to(2) \\ - \\ 2tv+2cb=2\to2\cdot(1) \\ _(------------------------) \\ 0tv+8cb=2 \end{gathered}`

we cleared cb

`\begin{gathered} 0tv+8cb=2 \\ 8cb=2 \\ cb=(2)/(8) \\ cb=(1)/(4) \end{gathered}`

It takes John 1/4 hour to repair a cable box, now we will transform it in minutes

`\begin{gathered} 1h\to60\min \\ (1)/(4)h\to X \\ X=((1)/(4)h\cdot60\min )/(1h) \\ X=(60)/(4)\min \\ X=15\min \end{gathered}`

John takes 15 minutes to repair a cable box, now we will transform it in minutes