Back to Tutor Intake Assessment: Understand that solutions to a system of two linear equations correspond to points of intersection of their graphs

8.EE.C.8.a Tutor Intake -- Systems of Equations, Graphical Interpretation

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Grade 8·8 problems·~8 min·Common Core Math - Grade 8·standard·8-ee-c-8a

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Concepts

What does it mean for an ordered pair $(x, y)$ to be "the
solution" to a system of two linear equations?

It is a single ordered pair that makes BOTH equations true at the same time.

It is two separate answers — one $x$ -value from the first equation and one $y$ -value from the second.

It is any point that makes at least one of the two equations true.

It is the point where either line crosses the x-axis.

Line A has slope 2.001. Line B has slope 2. On a typical graph
they look almost parallel and very close together. Does this
system have a solution?

Yes — since the slopes are different (2.001 ≠ 2), the lines must intersect at exactly one point, even if that point is far from the visible graph area.

No — the lines look so close together that they must be considered parallel with no solution.

Yes, infinitely many — lines that look this close together are essentially the same line.

It cannot be determined without graphing.

Procedures

Consider the system $y = 2x + 1$ and $y = -x + 7$ . Substitute
$x = 2$ into the first equation, $y = 2x + 1$ , and compute $y$ .

For the same system, $y = 2x + 1$ and $y = -x + 7$ , does the
point $(2, 5)$ satisfy the SECOND equation, $y = -x + 7$ ? Enter 1
for yes or 0 for no.

Without graphing: the system $y = 3x + 2$ and $y = 3x - 4$ has
lines with the same slope (3) but different y-intercepts (2 and
-4). How many solutions does this system have? Enter a number
(enter -1 to mean "infinitely many").

The system $y = x + 2$ and $2y = 2x + 4$ has a second equation
that simplifies to the same line as the first (divide both sides
of $2y = 2x + 4$ by 2 to see this). How many solutions does this
system have? Enter a number (enter -1 to mean "infinitely
many").

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