Solving linear systems by equating coefficients

1. solve both equations for y

Solve both equations for one variable. In this case, $yy$ is already singled out, so it is easier to solve for $yy$.

2. Set equal

Equate ${\mathrm{I}}^{\mathrm{\prime }}\backslash mathrm\{I\text{'}\}$ and $\mathrm{I}{\mathrm{I}}^{\mathrm{\prime }}\backslash mathrm\{II\text{'}\}$ .

$\Rightarrow 1.5x+2=0.5x+2.5\backslash Rightarrow1.5x+2=0.5x+2.5$

3. Solve equation for x

4. Plug in x to find y

Plug $xx$ into ${\mathrm{I}}^{\mathrm{\prime }}\backslash mathrm\{I\text{'}\}$ or $\mathrm{I}{\mathrm{I}}^{\mathrm{\prime }}\backslash mathrm\{II\text{'}\}$ .

$y=0.5\cdot 0.5+2.5=0.25+2.5=2.75y\; =\; 0.5\; \backslash cdot\; 0.5\; +\; 2.5\; \backslash \backslash =\; 0.25\; +\; 2.5\; \backslash \backslash =\; 2.75$

Write down the solution set.

${\displaystyle L=\left\{(0.5;2.75)\right\}}\backslash displaystyle\{L\; =\; \backslash left\backslash lbrace\; \backslash left(0.5\; \backslash \; ;\; \backslash \; 2.75\; \backslash right)\; \backslash right\backslash rbrace\}$

Solving linear systems by equating coefficients

1. Solve both equations for x

Solve both equations for one variable. For example, for $xx$.

2. Set equal

Set equal ${\mathrm{I}}^{\mathrm{\prime }\mathrm{\prime }}\backslash mathrm\{I\text{'}\text{'}\}$ and $\mathrm{I}{\mathrm{I}}^{\mathrm{\prime }\mathrm{\prime }}\backslash mathrm\{II\text{'}\text{'}\}$ .

$\Rightarrow 0.5y-2=0.4y-0.2\backslash Rightarrow\; 0.5y-2\; =\; 0.4y-0.2$

3. Solve for y

4. Plug in y to find x

plug $yy$ into ${\mathrm{I}}^{\mathrm{\prime }\mathrm{\prime }}\backslash mathrm\{I\text{'}\text{'}\}$

$0.5\cdot 18-2=x=9-2=70.5\backslash cdot\; 18\; -2\; =\; x\; =\; 9-2\; =\; 7$

Write down the solution set

$L=\{(\mathrm{7,18})\}L\; =\; \backslash \{(7\{,\}18)\backslash \}$

Solving linear systems by equating coefficients

$\phantom{\mathrm{I}}\mathrm{I})2x+3y=4x-5\backslash hphantom\{\backslash mathrm\{I\}\}\backslash mathrm\{I\})\; \backslash quad\; 2x+3y\; =\; 4x\; -5$

$\mathrm{I}\mathrm{I})3x-2y=2y+8\backslash mathrm\{II\})\; \backslash quad\; 3x-2y\; =\; 2y+8$

1. Solve both equations for one variable (for instance y)

2. Set equal

Set equal ${\mathrm{I}}^{\mathrm{\prime }\mathrm{\prime }}\backslash mathrm\{I\text{'}\text{'}\}$ and $\mathrm{I}{\mathrm{I}}^{\mathrm{\prime }\mathrm{\prime }}\backslash mathrm\{II\text{'}\text{'}\}$ .

3. Solve for that one variable

Solve for $xx$.

4. Plug into one equation to find the other variable

Plug $xx$ into $\mathrm{I}{\mathrm{I}}^{\mathrm{\prime }}\backslash mathrm\{II\text{'}\}$.

Now, you can write down the solution set again

$L=\{(\mathrm{4,1})\}L\; =\; \backslash \{(4\{,\}1)\backslash \}$