Question

Solve the linear system using all three methods. $$ \begin{aligned} &x+y=2\\\ &6 x+y=2 \end{aligned} $$

Short answer

The solution to the system of equations is \(x = 0, y = 2\).

Step-by-step solution

Substitution Method - Step 1: Rearrange one equation

Let's start by rearranging the first equation \(x+y=2\) to isolate one variable. Let \(x = 2 - y\).

Substitution Method - Step 2: Substitute into the second equation

Next, substitute \(x\) in the second equation \(6x + y = 2\) with the expression obtained in step 1, leading to \(6(2 - y) + y = 2\).

Substitution Method - Step 3: Solve for \(y\)

Solve the equation \(12 - 6y + y = 2\), which simplifies to \(5y = 10\). Solving it gives \(y = 2\).

Substitution Method - Step 4: Solve for \(x\)

Substitute \(y = 2\) into the expression for \(x\) obtained in step 1 to get \(x = 2 - 2 = 0\). So, the solution is \(x = 0, y = 2\).

Elimination Method - Step 1: Subtract second equation from the first

(x + y) - (6x + y) = 2 - 2. This simplifies to -5x = 0, and solving for \(x\) gives \(x = 0\).

Elimination Method - Step 2: Substitute \(x = 0\) into the first equation

Substituting \(x = 0\) into the first equation \(x + y = 2\) gives \(y = 2\). So, the solution is \(x = 0, y = 2\).

Graphing Method - Step 1: Rewrite equations in slope-intercept form

The equations \(x + y = 2\) and \(6x + y = 2\) rewritten in \(y = mx + c\) format are \(y = -x + 2\) and \(y = -6x + 2\) respectively.

Graphing Method - Step 2: Find intersection

By graphing the equations on a coordinate plane, identify their point of intersection. This is the solution to the system. Here, the two lines intersect at the point (0,2). So, the solution is \(x = 0, y = 2\).