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) $\quad 8 x+6 y=-15$ $10 x-6 y=16$ $$ \begin{aligned} &y=4 x-7\\ &6 x+3 y=19 \end{aligned} $$

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) $\quad 8 x+6 y=-15$
$10 x-6 y=16$
$$
\begin{aligned}
&y=4 x-7\\
&6 x+3 y=19
\end{aligned}
$$

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Key Concepts

Systems of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. The goal is to find a common solution that satisfies all equations simultaneously, which often represents the point of intersection of lines in the case of two variables.

Substitution Method

The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation. This method is particularly useful when one equation is already solved or can be easily solved for one variable, reducing the system to a single equation in one variable.

Elimination Method

The elimination method focuses on aligning coefficients and adding or subtracting equations to cancel out one of the variables. By manipulating the equations to eliminate one variable, the system can be reduced to a single equation which is then solved for the remaining variable.

Algebraic Simplification

Algebraic simplification involves applying fundamental algebraic operations such as distribution, combining like terms, and isolating variables. These steps are crucial for transforming equations into simpler equivalent forms that are easier to solve.

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