Question

What is an equation of the line that passes through the points in the table?

• A. $\mathrm{y}=\frac{1}{3}\mathrm{x}+1$
• B. $\mathrm{y}=4\mathrm{x}$
• C. $\mathrm{y}=3\mathrm{x}+1$
• D. $\mathrm{y}=\frac{1}{4}\mathrm{x}$

Answer 1

The correct option is C

$\mathrm{y}=3\mathrm{x}+1$

Step 1: Check option A.

Given: The points in the table are $(1,4),(2,7),(3,10)$and $(4,13)$.

We know that to find the equation of the line we need to check whether all points are solutions of it.

Option A: $\mathrm{y}=\frac{1}{3}\mathrm{x}+1$

for point $(1,4)$:

$$\begin{gathered} \mathrm{L}.\mathrm{H}.\mathrm{S}:\mathrm{y}=4 \\ \mathrm{R}.\mathrm{H}.\mathrm{S}:\frac{1}{3}\mathrm{x}+1=\frac{1}{3}\left(1\right)+1=\frac{1}{3}+1=\frac{1+3}{3}=\frac{4}{3} \end{gathered}$$

for point $(2,7)$:

$$\begin{gathered} \mathrm{L}.\mathrm{H}.\mathrm{S}:\mathrm{y}=7 \\ \mathrm{R}.\mathrm{H}.\mathrm{S}:\frac{1}{3}\mathrm{x}+1=\frac{1}{3}\left(2\right)+1=\frac{2}{3}+1=\frac{2+3}{3}=\frac{5}{3} \end{gathered}$$

for point $(3,10)$:

$$\begin{gathered} \mathrm{L}.\mathrm{H}.\mathrm{S}:\mathrm{y}=10 \\ \mathrm{R}.\mathrm{H}.\mathrm{S}:\frac{1}{3}\mathrm{x}+1=\frac{1}{3}\left(3\right)+1=\frac{3}{3}+1=1+1=2 \end{gathered}$$

for point $(4,13)$:

$$\begin{gathered} \mathrm{L}.\mathrm{H}.\mathrm{S}:\mathrm{y}=13 \\ \mathrm{R}.\mathrm{H}.\mathrm{S}:\frac{1}{3}\mathrm{x}+1=\frac{1}{3}\left(4\right)+1=\frac{4}{3}+1=\frac{4+3}{3}=\frac{7}{3} \end{gathered}$$

No point satisfies the equation, hence this is not the required equation.

Step 2: Check option B:

Option B: $\mathrm{y}=4\mathrm{x}$

for point $(1,4)$:

$$\begin{gathered} \mathrm{L}.\mathrm{H}.\mathrm{S}:\mathrm{y}=4 \\ \mathrm{R}.\mathrm{H}.\mathrm{S}:4\mathrm{x}=4\left(1\right)=4 \end{gathered}$$

for point $(2,7)$:

$$\begin{gathered} \mathrm{L}.\mathrm{H}.\mathrm{S}:\mathrm{y}=7 \\ \mathrm{R}.\mathrm{H}.\mathrm{S}:4\mathrm{x}=4\left(2\right)=8 \end{gathered}$$

for point $(3,10)$:

$$\begin{gathered} \mathrm{L}.\mathrm{H}.\mathrm{S}:\mathrm{y}=10 \\ \mathrm{R}.\mathrm{H}.\mathrm{S}:4\mathrm{x}=4\left(3\right)=12 \end{gathered}$$

for point $(4,13)$:

$$\begin{gathered} \mathrm{L}.\mathrm{H}.\mathrm{S}:\mathrm{y}=13 \\ \mathrm{R}.\mathrm{H}.\mathrm{S}:4\mathrm{x}=4\left(4\right)=16 \end{gathered}$$

Only one point satisfies the equation, hence it is not the required equation.

Step 3: Check option C.

Option C: $\mathrm{y}=3\mathrm{x}+1$
for point $(1,4)$:

$$\begin{gathered} \mathrm{L}.\mathrm{H}.\mathrm{S}:\mathrm{y}=4 \\ \mathrm{R}.\mathrm{H}.\mathrm{S}:3\mathrm{x}+1=3\left(1\right)+1=3+1=4 \end{gathered}$$

for point $(2,7)$:

$$\begin{gathered} \mathrm{L}.\mathrm{H}.\mathrm{S}:\mathrm{y}=7 \\ \mathrm{R}.\mathrm{H}.\mathrm{S}:3\mathrm{x}+1=3\left(2\right)+1=6+1=7 \end{gathered}$$

for point $(3,10)$:

$$\begin{gathered} \mathrm{L}.\mathrm{H}.\mathrm{S}:\mathrm{y}=10 \\ \mathrm{R}.\mathrm{H}.\mathrm{S}:3\mathrm{x}+1=3\left(3\right)+1=9+1=10 \end{gathered}$$

for point $(4,13)$:

$$\begin{gathered} \mathrm{L}.\mathrm{H}.\mathrm{S}:\mathrm{y}=13 \\ \mathrm{R}.\mathrm{H}.\mathrm{S}:3\mathrm{x}+1=3\left(4\right)+1=12+1=13 \end{gathered}$$

All points satisfy the given equation. It is the required equation.

Step 4: Check option D.

Option D: $\mathrm{y}=\frac{1}{4}\mathrm{x}$

for point $(1,4)$:

$$\begin{gathered} \mathrm{L}.\mathrm{H}.\mathrm{S}:\mathrm{y}=4 \\ \mathrm{R}.\mathrm{H}.\mathrm{S}:\frac{1}{4}\mathrm{x}=\frac{1}{4}\left(1\right)=\frac{1}{4} \end{gathered}$$

for point $(2,7)$:

$$\begin{gathered} \mathrm{L}.\mathrm{H}.\mathrm{S}:\mathrm{y}=7 \\ \mathrm{R}.\mathrm{H}.\mathrm{S}:\frac{1}{4}\mathrm{x}=\frac{1}{4}\left(2\right)=\frac{2}{4}=\frac{1}{2} \end{gathered}$$

for point $(3,10)$:

$$\begin{gathered} \mathrm{L}.\mathrm{H}.\mathrm{S}:\mathrm{y}=10 \\ \mathrm{R}.\mathrm{H}.\mathrm{S}:\frac{1}{4}\mathrm{x}=\frac{1}{4}\left(3\right)=\frac{3}{4} \end{gathered}$$

for point $(4,13)$:

$$\begin{gathered} \mathrm{L}.\mathrm{H}.\mathrm{S}:\mathrm{y}=13 \\ \mathrm{R}.\mathrm{H}.\mathrm{S}:\frac{1}{4}\mathrm{x}=\frac{1}{4}\left(4\right)=\frac{4}{4}=1 \end{gathered}$$

No point satisfies the equation, hence this is not the required equation.

Since, the equation of line is $\mathrm{y}=3\mathrm{x}+1$.

Hence, the option C is the correct answer.