Question

topic essential question
what does it mean to solve a system of linear equations?
vocabulary review
complete each definition and then provide an example of each vocabulary word.
vocabulary
solution of a system of linear equations
system of linear equations
definition
example
any ordered pair that makes all equations in the
system true is a
.
a
is
formed by two or more linear equations that
use the same variables.
use vocabulary in writing
describe how you can find the number of solutions of two or more equations by
using the slope and the y-intercept. use vocabulary terms in your description.

Explanation:

For the first blank (definition of "solution of a system of linear equations"), the term is "solution of a system of linear equations" as the definition matches: an ordered pair making all equations in the system true. For the second blank (definition of "system of linear equations"), the term is "system of linear equations" since it's formed by two or more linear equations with the same variables. Examples: For "solution of a system of linear equations", take the system \( y = x + 1 \) and \( y = 2x - 1 \); solving, \( x + 1 = 2x - 1 \) gives \( x = 2 \), \( y = 3 \), so \((2, 3)\) is the solution. For "system of linear equations", an example is \( y = 3x + 2 \) and \( y = -x + 6 \).

Answer:

First blank: solution of a system of linear equations
Second blank: system of linear equations

Example for "solution of a system of linear equations": Consider the system \(

$$\begin{cases} y = x + 1 \\ y = 2x - 1 \end{cases}$$

\). Solving, substitute \( y \): \( x + 1 = 2x - 1 \), so \( x = 2 \), \( y = 3 \). The ordered pair \((2, 3)\) is the solution as it satisfies both equations.

Example for "system of linear equations": \(

$$\begin{cases} y = 3x + 2 \\ y = -x + 6 \end{cases}$$

\) (two linear equations with the same variables \( x, y \)).