Key Points

An equation of the form $ax+b=0$, where $a$ and $b$ are real numbers and $a eq 0$, is a linear equation in one variable. It has a unique (one and only one) solution.

Any equation that can be put in the form $ax+by+c=0$, where $a, b,$ and $c$ are real numbers, and $a$ and $b$ are not both zero, is called a linear equation in two variables.

The standard form is $ax+by+c=0$. To find the coefficients $a, b,$ and $c$, rearrange the given equation into this form. For example, in $2x = y$, the standard form is $2x - 1y + 0 = 0$, so $a=2, b=-1,$ and $c=0$.

A solution to a linear equation in two variables is a pair of values, one for $x$ and one for $y$, that satisfies the equation. It is written as an ordered pair $(x, y)$.

Unlike a linear equation in one variable, a linear equation in two variables has infinitely many solutions. For every chosen value of one variable, a corresponding value for the other variable can be found.

To find a solution, you can pick any value for one variable (e.g., $x$) and substitute it into the equation. Then, solve the resulting linear equation for the other variable (e.g., $y$).

An easy method to find two distinct solutions is to first set $x=0$ and solve for $y$, then set $y=0$ and solve for $x$. This gives the points where the line crosses the y-axis and x-axis.

To check if an ordered pair like $(p, q)$ is a solution to an equation, substitute $x=p$ and $y=q$. If the equation remains true (left side equals right side), then $(p, q)$ is a solution.

To represent a real-world statement, identify two unknown quantities, assign variables (like $x$ and $y$) to them, and write the relationship between them as an equation. For example, 'The cost of a notebook ($x$) is twice the cost of a pen ($y$)' is written as $x=2y$.

An equation in one variable, such as $ax+c=0$, can be written in the standard form for two variables as $ax+0y+c=0$. For example, $x=-5$ can be written as $1x+0y+5=0$.

An equation of the form $x=k$ is a linear equation in two variables. It can be written as $1x + 0y - k = 0$. Any point of the form $(k, y)$ is a solution, where $y$ can be any real number.

An equation of the form $y=k$ is a linear equation in two variables. It can be written as $0x + 1y - k = 0$. Any point of the form $(x, k)$ is a solution, where $x$ can be any real number.

If a specific point $(p, q)$ is given as a solution to an equation with an unknown constant (like $k$), you can find the constant by substituting $x=p$ and $y=q$ into the equation and solving for $k$. For example, if $(2,1)$ is a solution for $2x+3y=k$, then $2(2)+3(1)=k$, which gives $k=7$.