Question

Dear mathematician, I need help solving this equation. Please help me, thankyou.

Answer 1

The form of the linear equation is

`y=mx+b`

m is the slope

b is the y-intercept

For the 1st equation on the left

m = 1

b = -3

Substitute them in the form above

`y=x-3`

Add 3 to both sides

`\begin{gathered} y+3=x-3+3 \\ y+3=x \end{gathered}`

Subtract y from both sides

`\begin{gathered} y-y+3=x-y \\ 3=x-y \\ x-y=3 \end{gathered}`

This is the 4th equation on the right side

`m=1,b=-3\rightarrow x-y=3`

For the 2nd equation on the left

m = 1 and the line passes through the point (-1, 2)

Substitute m by 1 in the form

`y=x+b`

To find b substitute x by -1 and y by 2

`2=-1+b`

Add 1 to each side

`\begin{gathered} 2+1=-1+1+b \\ 3=b \end{gathered}`

Substitute it in the equation

`y=x+3`

Subtract 3 from each side

`\begin{gathered} y-3=x+3-3 \\ y-3=x \end{gathered}`

Subtract y from each side

`\begin{gathered} y-y-3=x-y \\ -3=x-y \\ x-y=-3 \end{gathered}`

This is the 3rd equation on the right side

`m=1,(-1,2)\rightarrow x-y=-3`

For the 3rd equation on the left

The line passes through points (-2, 3), (-3, 4)

We will use the rule of the slope to find m

`\begin{gathered} m=(y_2-y_1)/(x_2-x_1) \\ \\ x_1=-2,x_2=-3 \\ y_1=3,y_2=4 \\ \\ m=(4-3)/(-3--2)=(1)/(-3+2)=(1)/(-1)=-1 \end{gathered}`

Substitute it in the form of the equation

`y=-x+b`

To find b use one of the two given points

I will use the first point (-2, 3)

Substitute x by -2 and y by 3 in the equation to find b

`\begin{gathered} 3=-(-2)+b \\ 3=2+b \end{gathered}`

Subtract 2 from both sides

`\begin{gathered} 3-2=2-2+b \\ 1=b \end{gathered}`

Substitute it in the equation

`y=-x+1`

Add x to both sides

`\begin{gathered} x+y=-x+x+1 \\ x+y=1 \end{gathered}`

This is the 1st equation on the right side

`(-2,3),(-3,4)\rightarrow x+y=1`