Match the system listed with the choice from the column on...

Match the system listed with the choice from the column on the right that would be a subsequent step in solving the system. a) $-5 x+10 y=-15$ $5 x+3 y=4$ b) The lines intersect at $(0,-1)$ c) $6 x+3(4 x-7)=19$ d) $8 x=7$ e) $2 x-(5 x+3)=8$ f) $\quad 8 x+6 y=-15$ $10 x-6 y=16$ $$ \begin{array}{c} {x-2 y=3} \\ {5 x+3 y=4} \end{array} $$

Match the system listed with the choice from the column on the right that would be a subsequent step in solving the system.
a) $-5 x+10 y=-15$ $5 x+3 y=4$
b) The lines intersect at $(0,-1)$
c) $6 x+3(4 x-7)=19$
d) $8 x=7$
e) $2 x-(5 x+3)=8$
f) $\quad 8 x+6 y=-15$
$10 x-6 y=16$
$$
\begin{array}{c}
{x-2 y=3} \\
{5 x+3 y=4}
\end{array}
$$

Key Concepts

Algebraic Manipulation

This encompasses all the techniques such as distributing, factoring, multiplying, and dividing both sides of an equation by a nonzero constant. Mastery of these operations is essential for transforming the equations into a form where either the substitution or elimination method can be applied effectively.

Elimination Method

The elimination method is used to eliminate one of the variables by adding or subtracting the equations after appropriately scaling them. This simplifies the system to one equation with one variable. It is especially useful when the coefficients of one variable are opposites or can be made opposites through multiplication.

Simplification and Combining Like Terms

Before applying any solving method, it is often necessary to simplify the equations. This involves expanding brackets, combining like terms, and reducing the equations into a more manageable form. Simplification is a crucial step that makes further algebraic manipulations clearer and prevents errors in subsequent steps.

Systems of Linear Equations

This concept refers to a set of two or more linear equations involving the same set of variables. Understanding the structure of these systems is essential, as it allows one to apply appropriate solving techniques and to determine whether the system has a unique solution, infinitely many solutions, or no solution at all.

Substitution Method

The substitution method involves solving one of the equations for one variable in terms of the others, and then substituting this expression into the other equation(s). It is particularly useful when one equation is easily solvable for a variable, thereby reducing the system to a single equation with one unknown.

Transcript

Please give Ace some feedback