সাধারণ রূপ: x ও y অক্ষে লেখচিত্র অঙ্কন

Connecting the Dots

Remember that a standard form equation appears as: $$Ax + By = C,$$ where $A$, $B$, and $C$ aren't fractions, $A$ is positive, and $x$ and $y$ are variables.

Standard form does not give as much information about a graph as slope intercept form or point slope form, but it does make it fairly easy to find the $x$- and $y$-intercepts.

To draw a linear graph using standard form, follow the following steps:

1. Find the $x$-intercept by setting $y$ to $0$.
2. Find the $y$-intercept by setting $x$ to $0$.
3. Connect the dots.

Step 1: Find the $x$-intercept

Let's look at the equation: $$2x + 3y = 6$$ In order to easily graph this, we first need to find the $x$-intercept. To do that, we find $x$ when $y=0$.

When $y = 0$, $x$ is $3$. We can find this by taking the following steps: $$\begin{align} 2x + 3y &= 6\\ 2x + 3({\color{#8925ae}{0}}) &= 6\\ 2x&=6\\ \frac{2x}{\color{#8925ae}{2}}&=\frac{6}{\color{#8925ae}{2}}\\ x&=3 \end{align}$$ This gives us our $x$-intercept, $(3,0)$.

Step 2: Find the $y$-intercept

Next, we need to find the $y$-intercept. To do that, we need to find $y$ when $x = 0$.

When $y = 0$, $x$ is $3$. $$\begin{align} 2x+3y&=6\\ 2({\color{#8925ae}{0}})+3y&=6\\ 3y&=6\\ \frac{3y}{\color{#8925ae}{3}}&=\frac{6}{\color{#8925ae}{3}}\\ y&=2 \end{align}$$ This gives our $y$-intercept, $(0,2)$.

Step 3: Connect the Dots

With both intercepts plotted, we can graph the line by connecting the dots!

(Video) Graphing Linear Functions by Finding the $x$- and $y$-Intercepts

by PatrickJMT

This video by Patrick JMT explains how to draw a graph of an equation in standard form using the $x$- and $y$-intercepts.

$\large{\text{Summary}}$

Intercepts

An intercept is the point where the graph crosses an axis. So the $x$-intercept is the point where the graph crosses the $x$-axis and the $y$-intercept is the point where the graph crosses the $y$-axis.

If we look at a graph, we see that any $x$-intercept will have a $y$ coordinate of $0$. Similarly, any $y$-intercept will have an $x$ coordinate of $0$.

Using the standard form of an equation we can easily find the intercepts.

• To find the $x$-intercept, we set $y=0$ and solve for $x$.
• To find the $y$-intercept, we set $x=0$ and solve for $y$.

Then, because two points define a line, we can graph our equation by drawing a line through these intercepts!

Example

Plot the intercepts of $2x+4y=12$ and use them to graph the equation.

Starting with finding the $x$-intercept, we set $y=0$ and solve for $x$. $$\begin{align} 2x+4y&=12\\ 2x+4({\color{#8925ae}{0}})&=12\\ 2x&=12\\ \frac{2x}{\color{#8925ae}{2}}&=\frac{12}{\color{#8925ae}{2}}\\ x&=6 \end{align}$$ Our $x$-intercept is $(6,0)$.

For the $y$-intercept, we set $x=0$ and solve for $y$. $$\begin{align} 2x+4y&=12\\ 2({\color{#8925ae}{0}})+4y&=12\\ 4y&=12\\ \frac{4y}{\color{#8925ae}{4}}&=\frac{12}{\color{#8925ae}{4}}\\ y&=3 \end{align}$$ Our $y$-intercept is $(0,3)$.

Finally, we can plot these points, and draw the line that goes through them. This will be the graph of our equation!

Solution

To find the $x$-intercept, we let $y=0$ and solve for $x$. $$\begin{align} 10x+5y&=105\\ 10x+5({\color{#8925ae}{0}})&=105\\ 10x&=105\\ x&=\frac{21}5 \end{align}$$ So the $x$-intercept is at the point $(\frac{21}{2},0)$.

To find the $y$-intercept, we let $x=0$ and solve for $y$. $$\begin{align} 10x+5y&=105\\ 10({\color{#8925ae}{0}})+5y&=105\\ 5y&=105\\ y&=21 \end{align}$$ So the $y$-intercept is at the point $(0,21)$.

Graphing Linear Functions Using Intercepts

Another method to graph linear functions is to first plot the intercepts, the connect the dots.