Solutions to Linear Equations
The goal of this lesson and the next is to start getting students to think about linear equations in two variables in a different way in preparation for their work on systems of linear equations in the next unit. Until now, students have mostly been working with contexts where one variable depends on another, for example, distance depending on time. The linear equation representing such a situation is often written in the form \(y = mx + b\). In this lesson, they look at contexts where both variables have to satisfy a constraint, and a natural way to write the constraint is with an equation of the form \(Ax + By = C\). For example, the first activity gets students to think about different ways of spending a fixed sum of money on two differently priced items, and the second activity gets them to write an equation expressing a numerical constraint on two numbers (twice the first number plus the second number adds up to 10).
Pairs of numbers that make the equation true are solutions to the equation (with two variables); they are the coordinates of points that lie on the graph. Students also consider pairs of numbers that do not make the equation true, and notice that they do not lie on the graph. This insight is developed in the next lesson, where students look for points that are simultaneously the solution to two different equations.
- Comprehend that the points that lie on the graph of an equation represent exactly the solution set of the equation of the line (i.e., that every point on the line is a solution, and any point not on the line is not a solution).
- Create a graph and an equation in the form $Ax+By=C$ that represent a linear relationship.
- Determine pairs of values that satisfy or do not satisfy a linear relationship using an equation or graph.
Let’s think about what it means to be a solution to a linear equation with two variables in it.
- I know that the graph of an equation is a visual representation of all the solutions to the equation.
- I understand what the solution to an equation in two variables is.
solution to an equation with two variables
A solution to an equation with two variables is a pair of values of the variables that make the equation true.
For example, one possible solution to the equation \(4x+3y=24\) is \((6,0)\). Substituting 6 for \(x\) and 0 for \(y\) makes this equation true because \(4(6)+3(0)=24\).
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