A.3(B)·Algebra II·Systems of Equations & InequalitiesReadiness

Solving Systems

Use algebraic methods, graphs, and tables to solve systems of equations and inequalities.

Students solve three-variable linear systems by substitution or elimination and test ordered pairs against every inequality in a system.

Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.

Archived course: TEKS code and readiness label from TEA's 2013 Algebra II key. Algebra II is not one of Texas's current required STAAR EOC assessments, and this lesson does not change the Biology and Algebra I Study Plans.

Practice the released item Ask the tutor All Algebra II released practice50 minutes

Learn the concept

Solve a repeated sum

If two equations share a sum such as $y+z$ , substitute that sum as one quantity. This is equivalent to elimination and can isolate x immediately.

Recover and verify

After finding one variable, substitute into another equation, then find the third variable. A solution triple must make all three original equations true.

Intersect solution sets

For inequalities, substitute both coordinates into every condition. A linear and a quadratic inequality can share a solution region, but a point in only one region is not a system solution. Strict inequalities exclude equality.

Worked example

Solve $x+y+z=13$ , $x+2y=11$ , and $y+z=8$ .

Substitute $y+z=8$ into the first equation: $x+8=13$ , so $x=5$ .
Then $5+2y=11$ gives $y=3$ .
Use $3+z=8$ to get $z=5$ .
Check: $5+3+5=13$ , $5+2(3)=11$ , and $3+5=8$ .

$(x,y,z)=(5,3,5)$

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Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.

Texas public school video

Algebra 2 Solving Systems of Equations Using Graphs

UT IPSI · Kid2Kid / Texas Education Agency

Connect simultaneous solutions to shared graph points before algebraic and inequality checks.

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Try it yourself

TEKSForge practice · unscored

1. Find x in $x+y+z=17$ , $x+3y=19$ , $y+z=10$ .
Show answer and reasoning
$x=7$
The known sum gives $x+10=17$ . The remaining equations give $y=4,z=6$ , confirming consistency.
2. Which of $(1,2)$ and $(3,0)$ satisfies both $x+y<6$ and $x^2+y<7$ ?
Show answer and reasoning
Only $(1,2)$ .
Both pass the linear test. For the quadratic test, $1+2=3<7$ , but $9+0=9$ is not less than 7.

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Algebra II EOC · 2013 · Item 19 · 1 point

What is the $x$ -value of the solution to the system of equations $x + y + z = 8$ , $x + 2y = 6$ , and $y + z = 4$ ? Record your answer.

Show STAAR answer and explanation

Substitute $y+z=4$ into the first equation: $x+4=8$ , so $x=4$ . Then $4+2y=6$ gives $y=1$ , and $z=3$ . Check: $4+1+3=8$ .

1: 1 is the y-value of the solution; the question asks for x.

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Archived released STAAR Algebra II EOC · 2013 · Item 44 · 1 point

Which of the following ordered pairs is a solution to the system of inequalities $x + y < 4$ and $x^2 + y < 8$ ?

(1, 5)
(1, 2)
(4, −1)
(−4, −1)

Show STAAR answer and explanation

(1, 2)

For $(1,2)$ : $1+2=3<4$ and $1^2+2=3<8$ . It satisfies both inequalities.

(1, 5): $1+5=6$ is not less than 4.

(4, −1): $4^2+(-1)=15$ is not less than 8.

(−4, −1): $(-4)^2+(-1)=15$ is not less than 8, even though $x+y=-5<4$ .

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Start scored STAAR practice

Learning objectives

By the end of the lesson, students can:

Use a known sum to simplify a three-variable system.
Find and check an ordered triple.
Determine whether an ordered pair satisfies both a linear and a quadratic inequality.

Instructional sequence

1
Engage
5 min
Test whether a pair satisfies two conditions at once.
2
Direct instruction
10 min
Model substituting an entire known sum in a three-variable system.
3
Guided practice
13 min
Solve for the remaining variables and verify all equations.
4
Compare and verify
15 min
Test candidate pairs in a linear-quadratic inequality system.
5
Closure
7 min
State the requested coordinate and complete the exit ticket.

Misconceptions to address

“Satisfying one inequality is enough.”

Teacher move: A system means all conditions hold simultaneously.

“Report the first number found regardless of the question.”

Teacher move: Label every variable and reread which coordinate is requested.

Learn the concept

Solve a repeated sum

Recover and verify

Intersect solution sets

Worked example

Watch the lesson

Algebra 2 Solving Systems of Equations Using Graphs

Try it yourself

1. Find x in x+y+z=17x+y+z=17x+y+z=17, x+3y=19x+3y=19x+3y=19, y+z=10y+z=10y+z=10.

2. Which of (1,2)(1,2)(1,2) and (3,0)(3,0)(3,0) satisfies both x+y<6x+y<6x+y<6 and x2+y<7x^2+y<7x2+y<7?

Released STAAR practice

Archived released STAAR Algebra II EOC · 2013 · Item 19 · 1 point

Archived released STAAR Algebra II EOC · 2013 · Item 44 · 1 point

Learning objectives

Instructional sequence

Engage

Direct instruction

Guided practice

Compare and verify

Closure

Misconceptions to address

1. Find x in $x+y+z=17$ , $x+3y=19$ , $y+z=10$ .

2. Which of $(1,2)$ and $(3,0)$ satisfies both $x+y<6$ and $x^2+y<7$ ?