Converting between Slope-Intercept and Standard Form
Sometimes linear equations are expressed in forms other than Slope-Intercept Form.
For example, if the Falcon Football team sells hamburgers for $4 each and hot dogs for $2 each and they make a total of $24 in their first hour of business, the equation: \[4x+2y=24\]
can be written to model this situation using \(x\) to represent the number of hamburgers sold and \(y\) to represent the number of hot dogs sold. We would need to know more in order to find the number of each sold.
This equation is expressed in Standard Form where both variables are written on the same side of the equality.
The process of solving an equation with two or more variables for one variable is called isolating a variable or solving a literal equation.
Instead of resulting in a number, which often happens when solving equations, we obtain a new formula. This formula expresses the value of one variable in terms of the others. That is, it tells us how to get the value of one variable when we know the others.
One useful purpose of this skill is to graph lines. For example, \(3x-2y=6\) is the equation of a line given in Standard Form. We commonly use the Slope-Intercept Form, \(y=mx+b\), to graph from the equation of a line since the slope, \(m,\) and the \(y\)-intercept, \(b,\) are clearly visible in the equation.
To change the equation from Standard Form to Slope-Intercept Form, we must isolate the variable “\(y\)”, meaning that the equation is solved for \(y\) in terms of \(x\).
Investigate this idea to develop a process for isolating a variable.
- Use the equation \(3x-2y=6\).
- Substitute the value \(x=4\) into the equation \(3x-2y=6\). Isolate the variable \(y\) by solving the equation for \(y\).) Show your work for each step.
- Substitute the value \(x=-6\) into the equation \(3x-2y=6\). Isolate the variable \(y\) by solving the equation for \(y\). Show your work for each step.
- Now solve the equation \(3x-2y=6\) for \(y\) without knowing a value for \(x\). Leave your answer in terms of \(x\) which means that “\(x\)” will remain in your final equation.
- What is the slope of the line \(3x-2y=6\)?
- What is the \(y\)-intercept of the line \(3x-2y=6\)?
2. Solve the equation \(4x+2y=24\) for “\(y\)” in terms of “\(x\)” and identify the slope and the \(y\)-intercept.