Solutions to an equation with two variables
Find solutions to two-variable equations in Grade 9 algebra by substituting x-values to compute y-values, producing ordered pairs (x,y) that satisfy the equation and lie on its graph.
Key Concepts
Property A solution to an equation with two variables is an ordered pair $(x, y)$ that makes the equation true. Solutions can be found by substituting values for the independent variable, $x$, to find the corresponding value of the dependent variable, $y$.
Examples For the equation $y = 3x 1$, if $x=2$, then $y = 3(2) 1 = 5$. The solution is $(2, 5)$. For the equation $y = 3x 1$, if $x=0$, then $y = 3(0) 1 = 1$. The solution is $(0, 1)$. For the equation $y = 3x 1$, if $x= 3$, then $y = 3( 3) 1 = 10$. The solution is $( 3, 10)$.
Explanation Think of an equation like $y = 3x 1$ as a magic function machine. You put a number for $x$ in, the machine does its magic (multiplies by 3, subtracts 1), and spits out the corresponding $y$. The input and output pair is a perfect 'solution'!
Common Questions
What is a solution to an equation with two variables?
A solution is an ordered pair (x, y) that makes the equation true when both values are substituted. For y = 2x + 3, the pair (1, 5) is a solution because 5 = 2(1) + 3. There are infinitely many solutions.
How do you find solutions to a two-variable equation?
Choose any value for x, substitute it into the equation, and solve for y. Repeat with different x-values to generate a table of ordered pairs. Each pair represents a point on the graph of the equation.
How do solutions to a two-variable equation relate to its graph?
Every solution (x, y) corresponds to a point on the graph of the equation. For linear equations, the graph is a straight line, and every point on that line is a solution to the equation.