### Clarifications

*Clarification 1*: Within this benchmark, the expectation is to solve systems using elimination, substitution and graphing.

*Clarification 2*: Within the Algebra 1 course, the system is limited to two equations.

**Subject Area:**Mathematics (B.E.S.T.)

**Grade:**912

**Strand:**Algebraic Reasoning

**Date Adopted or Revised:**08/20

**Status:**State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

### Terms from the K-12 Glossary

- Linear Equation

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 8, students determined whether a system of linear equations had one solution, no solution or infinitely many solutions and solved such systems graphically. In Algebra I, students solve systems of linear equations in two variables algebraically and graphically. In later courses, students will solve systems of linear equations in three variables and systems of nonlinear equations in two variables.- For students to have full understanding of systems, instruction should include MA.912.AR.9.4 and MA.912.AR.9.6. Equations and inequalities and their constraints are all related and the connections between them should be reinforced throughout instruction.
- Instruction allows students to solve using any method (substitution, elimination or
graphing) but recognizing that one method may be more efficient than another
*(MTR.3.1)*.- If both equations are given in standard form, then elimination, or linear combination, may be most efficient.
- If one equation is given in slope-intercept form or solved for $x$, then substitution may be easiest.
- If both equations are given in standard form, then elimination, or linear combination, may be most efficient.

- Consider presenting a system that favors one of these methods and having students divide
into three groups to solve them using different methods. Have students share their work
and discuss which method was more efficient than the others
*(*.*MTR.3.1*, MTR.4.1) - Include cases where students must interpret solutions to systems of equations.
- Instruction includes the use of various forms of linear equations.
- Standard Form Can be described by the equation $A$$x$ + $B$$y$ = $c$, where $A$, $B$ and $C$ are any rational number.
- Slope-Intercept Form Can be described by the equation $y$ = $m$$x$ + $b$, where $m$ is the slope and $b$ is the $y$-intercept.
- Point-Slope Form
Can be described by the equation $y$ − $y$
_{1}= $m$($x$ − $x$_{1}), where ($x$_{1}, $y$_{1}) are a point on the line and $m$ is the slope of the line.

- When introducing the elimination method, students may express confusion when considering adding equations together. Historically, students have used the properties of equality to create equivalent equations to solve for a variable of interest. In most of these efforts, operations performed on both sides of the original equation have been identical. With the introduction of the elimination method, students can now see that operations performed on each side of an equation must be equivalent (not necessarily identical) for the property to hold. Guide students to explore forming equivalent equations with simpler equations by adding or subtracting equivalent values. Lead them to see that the new equations they generate have the same solutions. Have them discuss why the method works: equations are simply pairs of equivalent expressions, which is why they can be added/subtracted with each other.

### Common Misconceptions or Errors

- Students may not understand linear systems of equations can only have more than one solution if there are infinitely many solutions.
- Students may not understand linear systems of equations can have no solution.
- Students may have difficulty making connections between graphic and algebraic representations of systems of equations.
- Students may have difficulty choosing the best method of finding the solution to a system of equations.
- Students may have difficulty translating word problems into systems of equations and inequalities.
- Students using the elimination method may alter the original equations in a way that creates like terms that can be subtracted. When subtracting across the two equations students may have difficulty remembering to apply the subtraction to the remaining terms and constants.

### Strategies to Support Tiered Instruction

- Instruction includes opportunities to use graphing software to visualize the possible solutions for a system of equations. Systems of equations only produce three different types of solutions: one solution, infinite solutions, and no solutions. Each type of system can be graphed for analysis of each type of solution set.
- Teacher models through a think-aloud how a system of equations can have no solutions.
- For example, “I can algebraically solve a system with no solutions. The solution will reveal that the left and right sides of the equation cannot be equal, causing a no solution set. In addition, if I rearrange both equations to the slope-intercept form, the equations will have the same slope. I can utilize my knowledge of parallel lines to understand that the system cannot have any solutions.”

- Teacher provides step-by-step process for solving systems.
- For example, when solving the system below, students can use the method of
elimination.2$x$ + 4$y$ = −10

3$x$ + 5$y$ = 8

- For example, when solving the system below, students can use the method of
elimination.

- If the student chooses to eliminate the $y$-variable, they can multiply the first
equation by 5 and the second by 4 so that both coefficients of $y$ are 20. 5(2$x$ + 4$y$ = −10) to 10$x$ + 20$y$ = −50

4(3$x$ + 5$y$ = 8) to 12$x$ + 20$y$ = 32

- The student either subtract the two new equations, or creates additive inverses by
multiplying one of the equations by −1 (as shown) and then adds the equations. −1(10$x$ + 20$y$ = −50) to −10$x$ − 20$y$ = +50

−10$x$ − 20$y$ = +50__12$x$ + 20$y$ = 32__

2$x$ = 82

$x$ = 41

- Once students determine one of the values ($x$ in this case), then they can
substitute this back into one of the given equations to find the other value ($y$ in
this case). 2(41) + 4$y$ = −10

4$y$ = −10−82

$y$ = −23

### Instructional Tasks

*Instructional Task 1 (*

*MTR.3.1*,*MTR.4.1*)- You and a friend go to Tacos Galore for lunch. You order three soft tacos and three burritos
and your total bill is $11.25. Your friend’s bill is $10.00 for four soft tacos and two burritos.
- Part A. Write a system of two-variable linear equations to represent this situation.
- Part B. Solve the system both algebraically and graphically to determine the cost of each burrito and each soft taco.
- Part C. Is one method more efficient than the other? Why or why not?

Instructional Task 2 (

Instructional Task 2 (

*MTR.3.1*,*MTR.4.1*)- Part A. Determine the solution to the system of linear equations below using your method of
choice. 0.5$x$ − 1.4$y$ = 5.8

$y$ = −0.3$x$ − $\frac{\text{1}}{\text{5}}$

- Part B. Discuss with a partner why you chose that method.

### Instructional Items

*Instructional Item 1*

- Determine the exact solution of the system of linear equations below.−$\frac{\text{1}}{\text{10}}$$x$ + $\frac{\text{1}}{\text{2}}$$y$ = $\frac{\text{4}}{\text{5}}$

$\frac{\text{1}}{\text{7}}$$x$ + $\frac{\text{1}}{\text{3}}$$y$ = −$\frac{\text{2}}{\text{21}}$

*Instructional Item 2*

- Carla volunteered to make pies for a bake sale. She bought two pounds of apples and six pounds of peaches and spent $19. After baking the pies, she decided they looked so good she would make more. She went back to the store and bought another pound of apples and five more pounds of peaches and spent $15. Write a system of linear equations that describes her purchases, where $a$ represents the cost per pound of the apples and $p$ represents the cost per pound of the peaches.

**The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.*

## Related Courses

## Related Access Points

## Related Resources

## Formative Assessments

## Lesson Plans

## Original Student Tutorials

## Perspectives Video: Professional/Enthusiast

## Tutorials

## Video/Audio/Animations

## MFAS Formative Assessments

Students are asked to solve a system of equations with rational solutions either algebraically or by graphing and are asked to justify the choice of method.

Students are asked to determine the number of solutions of each of four systems of linear equations without solving the systems of equations.

Students are asked to solve a system of equations both algebraically and graphically.

Students are asked to solve a system of equations both algebraically and graphically.

Students are asked to solve a system of equations both algebraically and graphically.

Students are asked to solve three systems of linear equations algebraically.

Students are asked to solve a word problem by solving a system of linear equations.

Students are given word problems and asked to write a pair of simultaneous linear equations that could be used to solve them.

## Original Student Tutorials Mathematics - Grades 9-12

Follow as we learn why the *x*-coordinate of the point of intersection of two functions is the solution of the equation *f*(*x*) = *g*(*x*) in this interactive tutorial.

Learn how to solve systems of linear equations graphically in this interactive tutorial.

Learn to solve systems of linear equations using substitution in this interactive tutorial.

This part 2 in a 7-part series. Click below to explore the other tutorials in the series.

Part 1: Solving Systems of Linear Equations Part 1: Using Graphs

Part 3: Solving Systems of Linear Equations Part 3: Basic Elimination (Coming soon)

Part 4: Solving Systems of Linear Equations Part 4: Advanced Elimination (Coming soon)

Part 5: Solving Systems of Linear Equations Part 5: Connecting Algebraic Methods to Graphing (Coming soon)

Part 6: Solving Systems of Linear Equations Part 6: Writing Systems from Context (Coming soon)

Part 7: Solving Systems of Linear Equations Part 7: Word Problems (Coming soon)

Learn to solve systems of linear equations using basic elimination in this interactive tutorial.

This part 3 in a 7-part series. Click below to explore the other tutorials in the series.

Part 1: Solving Systems of Linear Equations Part 1: Using Graphs

Part 2: Solving Systems of Linear Equations Part 2: Substitution

Part 4: Solving Systems of Linear Equations Part 4: Advanced Elimination (Coming soon)

Part 5: Solving Systems of Linear Equations Part 5: Connecting Algebraic Methods to Graphing (Coming soon)

Part 6: Solving Systems of Linear Equations Part 6: Writing Systems from Context (Coming soon)

Part 7: Solving Systems of Linear Equations Part 7: Word Problems (Coming soon)

Learn to solve systems of linear equations using advanced elimination in this interactive tutorial.

This part 4 in a 7-part series. Click below to explore the other tutorials in the series.

**Part 1: Solving Systems of Linear Equations Part 1: Using Graphs****Part 2: Solving Systems of Linear Equations Part 2: Substitution****Part 3: Solving Systems of Linear Equations Part 3: Basic Elimination**- Part 5: Solving Systems of Linear Equations Part 5: Connecting Algebraic Methods to Graphing (Coming soon)
- Part 6: Solving Systems of Linear Equations Part 6: Writing Systems from Context (Coming soon)
- Part 7: Solving Systems of Linear Equations Part 7: Word Problems (Coming soon)

Learn to solve systems of linear equations by connecting algebraic and graphing methods in this interactive tutorial.

This part 5 in a 7-part series. Click below to explore the other tutorials in the series.

- Part 1: Solving Systems of Linear Equations: Using Graphs
- Part 2: Solving Systems of Linear Equations: Substitution
- Part 3: Solving Systems of Linear Equations: Basic Elimination
- Part 4: Solving Systems of Linear Equations: Advanced Elimination
- Part 6: Solving Systems of Linear Equations: Writing Systems from Context (Coming soon)
- Part 7: Solving Systems of Linear Equations: Word Problems (Coming soon)

Learn how to create systems of linear equations to represent contextual situations in this interactive tutorial.

This part 6 in a 7-part series. Click below to explore the other tutorials in the series.

- Part 1: Solving Systems of Linear Equations: Using Graphs
- Part 2: Solving Systems of Linear Equations: Substitution
- Part 3: Solving Systems of Linear Equations: Basic Elimination
- Part 4: Solving Systems of Linear Equations: Advanced Elimination
- Part 5: Solving Systems of Linear Equations: Connecting Algebraic Methods to Graphing
- Part 7: Solving Systems of Linear Equations: Word Problems (Coming soon)

Learn to solve word problems represented by systems of linear equations, algebraically and graphically, in this interactive tutorial.

This part 7 in a 7-part series. Click below to explore the other tutorials in the series.

- Part 1: Solving Systems of Linear Equations: Using Graphs
- Part 2: Solving Systems of Linear Equations: Substitution
- Part 3: Solving Systems of Linear Equations: Basic Elimination
- Part 4: Solving Systems of Linear Equations: Advanced Elimination
- Part 5: Solving Systems of Linear Equations: Connecting Algebraic Methods to Graphing
- Part 6: Solving Systems of Linear Equations: Writing Systems from Context

## Student Resources

## Original Student Tutorials

Learn to solve word problems represented by systems of linear equations, algebraically and graphically, in this interactive tutorial.

This part 7 in a 7-part series. Click below to explore the other tutorials in the series.

- Part 1: Solving Systems of Linear Equations: Using Graphs
- Part 2: Solving Systems of Linear Equations: Substitution
- Part 3: Solving Systems of Linear Equations: Basic Elimination
- Part 4: Solving Systems of Linear Equations: Advanced Elimination
- Part 5: Solving Systems of Linear Equations: Connecting Algebraic Methods to Graphing
- Part 6: Solving Systems of Linear Equations: Writing Systems from Context

Type: Original Student Tutorial

Learn how to create systems of linear equations to represent contextual situations in this interactive tutorial.

This part 6 in a 7-part series. Click below to explore the other tutorials in the series.

- Part 1: Solving Systems of Linear Equations: Using Graphs
- Part 2: Solving Systems of Linear Equations: Substitution
- Part 3: Solving Systems of Linear Equations: Basic Elimination
- Part 4: Solving Systems of Linear Equations: Advanced Elimination
- Part 5: Solving Systems of Linear Equations: Connecting Algebraic Methods to Graphing
- Part 7: Solving Systems of Linear Equations: Word Problems (Coming soon)

Type: Original Student Tutorial

Learn to solve systems of linear equations by connecting algebraic and graphing methods in this interactive tutorial.

This part 5 in a 7-part series. Click below to explore the other tutorials in the series.

- Part 1: Solving Systems of Linear Equations: Using Graphs
- Part 2: Solving Systems of Linear Equations: Substitution
- Part 3: Solving Systems of Linear Equations: Basic Elimination
- Part 4: Solving Systems of Linear Equations: Advanced Elimination
- Part 6: Solving Systems of Linear Equations: Writing Systems from Context (Coming soon)
- Part 7: Solving Systems of Linear Equations: Word Problems (Coming soon)

Type: Original Student Tutorial

Learn to solve systems of linear equations using advanced elimination in this interactive tutorial.

This part 4 in a 7-part series. Click below to explore the other tutorials in the series.

**Part 1: Solving Systems of Linear Equations Part 1: Using Graphs****Part 2: Solving Systems of Linear Equations Part 2: Substitution****Part 3: Solving Systems of Linear Equations Part 3: Basic Elimination**- Part 5: Solving Systems of Linear Equations Part 5: Connecting Algebraic Methods to Graphing (Coming soon)
- Part 6: Solving Systems of Linear Equations Part 6: Writing Systems from Context (Coming soon)
- Part 7: Solving Systems of Linear Equations Part 7: Word Problems (Coming soon)

Type: Original Student Tutorial

Learn to solve systems of linear equations using basic elimination in this interactive tutorial.

This part 3 in a 7-part series. Click below to explore the other tutorials in the series.

Part 1: Solving Systems of Linear Equations Part 1: Using Graphs

Part 2: Solving Systems of Linear Equations Part 2: Substitution

Part 4: Solving Systems of Linear Equations Part 4: Advanced Elimination (Coming soon)

Part 5: Solving Systems of Linear Equations Part 5: Connecting Algebraic Methods to Graphing (Coming soon)

Part 6: Solving Systems of Linear Equations Part 6: Writing Systems from Context (Coming soon)

Part 7: Solving Systems of Linear Equations Part 7: Word Problems (Coming soon)

Type: Original Student Tutorial

Learn to solve systems of linear equations using substitution in this interactive tutorial.

This part 2 in a 7-part series. Click below to explore the other tutorials in the series.

Part 1: Solving Systems of Linear Equations Part 1: Using Graphs

Part 3: Solving Systems of Linear Equations Part 3: Basic Elimination (Coming soon)

Part 4: Solving Systems of Linear Equations Part 4: Advanced Elimination (Coming soon)

Part 6: Solving Systems of Linear Equations Part 6: Writing Systems from Context (Coming soon)

Part 7: Solving Systems of Linear Equations Part 7: Word Problems (Coming soon)

Type: Original Student Tutorial

Learn how to solve systems of linear equations graphically in this interactive tutorial.

Type: Original Student Tutorial

Follow as we learn why the *x*-coordinate of the point of intersection of two functions is the solution of the equation *f*(*x*) = *g*(*x*) in this interactive tutorial.

Type: Original Student Tutorial

## Tutorials

This video is an example of solving a system of linear equations by elimination where the system has infinite solutions.

Type: Tutorial

This video shows how to solve a system of equations through simple elimination.

Type: Tutorial

This video explains how to identify systems of equations without a solution.

Type: Tutorial

This video shows how to solve systems of equations by elimination.

Type: Tutorial

This video is an introduction to the elimination method of solving a system of equations.

Type: Tutorial

This video demonstrates solving a word problem by creating a system of linear equations that represents the situation and solving them using elimination.

Type: Tutorial

In this tutorial, students will learn how to solve and graph a system of equations.

Type: Tutorial

This tutorial shows students how to solve a system of linear equations by graphing the two equations on the same coordinate plane and identifying the intersection point.

Type: Tutorial

This tutorial shows how to solve a system of equations by graphing. Students will see what a no solution system of equations looks like in a graph.

Type: Tutorial

This tutorial shows how to solve a system of equations using substitution.

Type: Tutorial

Systems of two linear equations in two variables can have a single solution, no solutions, or an infinite number of solutions. This video gives a great description of inconsistent, dependent, and independent systems. A consistent independent system of equations will have one solution. A consistent dependent system of equations will have infinite number of solutions, and an inconsistent system of equations will have no solution. This tutorial also provides information on how to distinguish a given system of linear equations as inconsistent, independent, or dependent system by looking at the slope and intercept.

Type: Tutorial

Systems of two equations in x and y can be solved by adding the equations to create a new equation with one variable eliminated. This new equation can then be solved to find the value of the remaining variable. That value is then substituted into either equation to find the value of other variable.

Type: Tutorial

A system of two equations in x and y can be solved by rearranging one equation to represent x in terms of y, and "substituting" this expression for x in the other equation. This creates an equation with only y which can then be solved to find y's value. This value can then be substituted into either equation to find the value of x.

Type: Tutorial

## Video/Audio/Animations

When should a system of equations with multiple variables be used to solve an Algebra problem, instead of using a single equation with a single variable?

Type: Video/Audio/Animation

This chapter presents a new look at the logic behind adding equations- the essential technique used when solving systems of equations by elimination.

Type: Video/Audio/Animation