Question

Is the slope different or the same?Is the Y-intercept different or the same? Is there infinitely many solutions?

Answer 1

Given the question in the question tab, the following are the solution steps to answer the question

STEP 1: Write the given equations

`\begin{gathered} y=x+1 \\ -x+y=3 \end{gathered}`

STEP 2: Write the equations in general slope-intercept form

`\begin{gathered} y=x+1---\text{ Line equation 1} \\ -x+y=3 \\ Add\text{ x to both sides to make y the subject of the formula} \\ -x+y+x=3+x \\ y=x+3---\text{Line equation }2 \end{gathered}`

STEP 3: Compare with the general slope-intercept form to get the slopes and the y-intercept

`\begin{gathered} y=mx+c \\ \text{where m is the slope, c is the y-intercept} \\ \text{For Line equation 1} \\ y=x+1 \\ m=1,c=1 \\ \text{For Line equation }2 \\ y=x+3 \\ m=1,c=3 \end{gathered}`

Hence, It can be seen from above that the slopes of the two equations are the same while the y-intercepts are different.

STEP 4: Define infinitely many solutions

An infinitely many solution has both sides equal. For example, 6x + 2=2+6x. If you simplify the equation using an infinite solutions formula or method, you'll get both sides equal, hence, it is an infinite solution. Infinite represents limitless or unboundedness. Infinite solutions would mean that any value for the variable would make the equation true.

From the definition above, it can be seen that there are no infinitely many solutions for the two eqautions.