Interpret the solution to a system of linear equations

Interpreting a system of linear equations means understanding its solution as the ordered pair that makes both equations true simultaneously, represented by the intersection of their graphs when the equations are graphed in the coordinate plane. The solution describes the values of two related quantities that satisfy both conditions; students distinguish one solution, no solution when lines are parallel, and infinitely many solutions when the equations represent the same line, rather than treating an intersection as valid without checking both equations.

Detailed Explanation: Interpret the solution to a system of linear equations

A solution to a system of linear equations is an ordered pair $(x,y)$ that makes both equations true at the same time. On a graph, it is the point where the two lines intersect.

Consider the system:

\begin{cases} y=2x+1\\ y=-x+7 \end{cases}

Step 1: Find where the equations have the same value

Both equations equal $y$ , so set their right sides equal:

2x+1=-x+7

Add $x$ to both sides:

3x+1=7

Subtract $1$ :

3x=6

Divide by $3$ :

x=2

Step 2: Find the corresponding $y$ -value

Substitute $x=2$ into either equation:

y=2(2)+1=5

So the solution is:

\boxed{(2,5)}

Step 3: Check the solution in both equations

First equation:

5=2(2)+1=5

Second equation:

5=-2+7=5

The ordered pair $(2,5)$ makes both equations true. Therefore, the graphs intersect at $(2,5)$ , and this point represents the values that satisfy both conditions.

A system can also have no solution if its lines are parallel, or infinitely many solutions if both equations describe the same line.

Learn by doing: Interpret the solution to a system of linear equations

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Types of Solutions - Graph to Zero, One, or Infinite

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