Task Sets: Prove system equivalence - Common Core Math - HS Algebra

Work through problems with immediate feedback

Recall / Warm-Up

Two systems of equations are called equivalent if they have:

The same number of equations

The same solution set — every solution of one is also a solution of the other

Equations that look the same (identical left and right sides)

The same coefficients on the $x$ -terms

The ordered pair $(2, 3)$ is a solution to a system if it satisfies:

At least one equation in the system

The first equation only

Every equation in the system simultaneously

The equation with the smallest coefficients

In solving the single-variable equation $2x + 4 = 10$ , we subtract 4 from both sides to get $2x = 6$ . This is justified by:

The Distributive Property

The Subtraction Property of Equality

The Multiplication Property of Equality

Simplification (combining like terms)

Fluency Practice

Consider the system:
$E_1$ : $2x + y = 7$
$E_2$ : $x - y = 2$

The REI.C.5 theorem says we may replace $E_1$ with $E_1 + k \cdot E_2$ for any constant $k$ . If $k = 1$ , what is the new equation that replaces $E_1$ ?

$2x + y + x - y = 7$ , simplifying to $3x = 7$

$2x + y + x - y = 7 + 2$ , simplifying to $3x = 9$

$2x + y = 7 + x - y$ , giving $x + 2y = 7$

$k(x - y) = k \cdot 2$ , giving $x - y = 2$

$A diagram showing two directions of the equivalence proof. Direction 1 arrows from the original system to the new system. Direction 2 arrows from the new system back to the original system.$

State the REI.C.5 theorem in your own words. Your statement should explain: (a) what operation is being performed on the system, and (b) what the theorem claims about the resulting system.

Prove Direction 1 of the REI.C.5 theorem: if $(x_0, y_0)$ is any solution of the original system ( $E_1 = 0$ and $E_2 = 0$ ), then $(x_0, y_0)$ also satisfies the new equation $E_1 + k \cdot E_2 = 0$ .

Write a complete proof with each step and its justification.

Which of the following operations on a system of two equations is legal (guaranteed to produce an equivalent system by the theorem)?

Replace $E_1$ with $E_1 + 5$ (add the constant 5 to the left side of $E_1$ )

Replace $E_1$ with $E_1 + (-2) \cdot E_2$

Replace $E_1$ with $\frac{E_1}{x}$ (divide $E_1$ by the variable $x$ )

Replace $E_1$ with $E_1^2$ (square both sides of $E_1$ )

A student wants to use the REI.C.5 theorem on the system:
$E_1$ : $3x + y = 10$
$E_2$ : $x + 2y = 8$

They choose $k = -3$ and form $E_1 + (-3) \cdot E_2$ . What does the new $E_1$ become?

$3x + y - 3x - 2y = 10 - 8$ , giving $-y = 2$

$3x + y - 3(x + 2y) = 10 - 3(8)$ , giving $-5y = -14$

$3x + y - 3x - 6y = 10 - 24$ , giving $-5y = -14$

$3x + y + (-3) = 10 + (-3)$ , giving $3x + y = 7$

You're viewing 2 of 6 sections.

Create a free account to continue the full exercise set and save your progress.

Create free account

0 of 8 answered

Answer all problems to submit.

Exercises: Prove That the Elimination Method Preserves Solution Sets

Recall / Warm-Up

Fluency Practice