Linear Equations in Two Variables

Is the following equation a linear equation in two variables? 1. \( y^2 = 2x + 3 \) 2. \( y - 2x = 9 \) The first equation, \( y^2 = 2x + 3 \), is not a linear equation in two variables because it contains \( y^2 \) which makes it a quadratic equation. On the other hand, the second equation, \( y - 2x = 9 \), is a linear equation in two variables. They both involve two variables, but the presence of \( y^2 \) in the first equation makes it non-linear.

Linear Equations in Two Variables

A linear equation in two variables is an equation that can be written in the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) and \( y \) are variables. The degree of the variables in a linear equation is always 1, which means they are not squared or raised to any higher power.

When determining whether an equation is linear in two variables, look for any terms that involve the variables raised to a power greater than 1. In the first equation \( y^2 = 2x + 3 \), the term \( y^2 \) makes it a non-linear equation. However, in the second equation \( y - 2x = 9 \), both \( y \) and \( x \) are raised to the power of 1, making it a linear equation in two variables.

Linear equations in two variables are especially useful in graphing and solving systems of equations. They represent lines in the coordinate plane and can be manipulated using algebraic techniques to find solutions for the variables.

In conclusion, the first equation is not a linear equation in two variables due to the presence of \( y^2 \), while the second equation is indeed a linear equation with both \( x \) and \( y \) raised to the power of 1.

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