Use simultaneous equations to find the point of intersection of the following pairs of lines: 4. $$ \begin{array}{l}x+y=5 \ 2 x-y=1\end{array} $$ 5. $$ \begin{array}{l}x-y=2 \ 2 x+y=7 \ 6 . \ 2 x+5 y=1 \ ext { 7. } \ ext { 7. } \ 2 x-3 y=4 \ 2 x+3 y=-8\end{array} $$ $$ \begin{array}{l}8 . \ 3 x-2 y=17 \ 9 .\end{array} $$ $$ 4 x+3 y=0 $$ $$ x+3 y=13 $$ $$ 2 x+5 y=21 $$

Question

Use simultaneous equations to find the point of intersection of the following pairs of lines: 4. $$ \begin{array}{l}x+y=5 \ 2 x-y=1\end{array} $$ 5. $$ \begin{array}{l}x-y=2 \ 2 x+y=7 \ 6 . \ 2 x+5 y=1 \ 	ext { 7. } \ 	ext { 7. } \ 2 x-3 y=4 \ 2 x+3 y=-8\end{array} $$ $$ \begin{array}{l}8 . \ 3 x-2 y=17 \ 9 .\end{array} $$ $$ 4 x+3 y=0 $$ $$ x+3 y=13 $$ $$ 2 x+5 y=21 $$

UpStudy AI · Accepted Answer

Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}x+y=5\2x-y=1\end{array}\right.$$
- step1: Solve the equation:
  $$\left\{ \begin{array}{l}x=5-y\2x-y=1\end{array}\right.$$
- step2: Substitute the value of $$x:$$
  $$2\left(5-y\right)-y=1$$
- step3: Simplify:
  $$10-3y=1$$
- step4: Move the constant to the right side:
  $$-3y=1-10$$
- step5: Subtract the numbers:
  $$-3y=-9$$
- step6: Change the signs:
  $$3y=9$$
- step7: Divide both sides:
  $$\frac{3y}{3}=\frac{9}{3}$$
- step8: Divide the numbers:
  $$y=3$$
- step9: Substitute the value of $$y:$$
  $$x=5-3$$
- step10: Calculate:
  $$x=2$$
- step11: Calculate:
  $$\left\{ \begin{array}{l}x=2\y=3\end{array}\right.$$
- step12: Check the solution:
  $$\left\{ \begin{array}{l}x=2\y=3\end{array}\right.$$
- step13: Rewrite:
  $$\left(x,y\right) = \left(2,3\right)$$
Solve the system of equations $$ x-y=2;2 x+y=7 $$.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}x-y=2\2x+y=7\end{array}\right.$$
- step1: Solve the equation:
  $$\left\{ \begin{array}{l}x=2+y\2x+y=7\end{array}\right.$$
- step2: Substitute the value of $$x:$$
  $$2\left(2+y\right)+y=7$$
- step3: Simplify:
  $$4+3y=7$$
- step4: Move the constant to the right side:
  $$3y=7-4$$
- step5: Subtract the numbers:
  $$3y=3$$
- step6: Divide both sides:
  $$\frac{3y}{3}=\frac{3}{3}$$
- step7: Divide the numbers:
  $$y=1$$
- step8: Substitute the value of $$y:$$
  $$x=2+1$$
- step9: Calculate:
  $$x=3$$
- step10: Calculate:
  $$\left\{ \begin{array}{l}x=3\y=1\end{array}\right.$$
- step11: Check the solution:
  $$\left\{ \begin{array}{l}x=3\y=1\end{array}\right.$$
- step12: Rewrite:
  $$\left(x,y\right) = \left(3,1\right)$$
Solve the system of equations $$ 3 x-2 y=17;4 x+3 y=0;x+3 y=13;2 x+5 y=21 $$.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}3x-2y=17\4x+3y=0\x+3y=13\2x+5y=21\end{array}\right.$$
- step1: Solve the equation:
  $$\left\{ \begin{array}{l}3x-2y=17\4x+3y=0\x=13-3y\2x+5y=21\end{array}\right.$$
- step2: Substitute the value of $$x:$$
  $$\left\{ \begin{array}{l}3\left(13-3y\right)-2y=17\4\left(13-3y\right)+3y=0\2\left(13-3y\right)+5y=21\end{array}\right.$$
- step3: Simplify:
  $$\left\{ \begin{array}{l}39-11y=17\52-9y=0\26-y=21\end{array}\right.$$
- step4: Solve the equation:
  $$\left\{ \begin{array}{l}y=2\52-9y=0\26-y=21\end{array}\right.$$
- step5: Substitute the value of $$y:$$
  $$\left\{ \begin{array}{l}52-9\times 2=0\26-2=21\end{array}\right.$$
- step6: Simplify:
  $$\left\{ \begin{array}{l}34=0\24=21\end{array}\right.$$
- step7: Calculate:
  $$\left\{ \begin{array}{l}x \in \varnothing \y \in \varnothing \end{array}\right.$$
- step8: Rewrite:
  $$(x, y) \in \varnothing$$
Solve the system of equations $$ 2 x+5 y=1;2 x-3 y=4;2 x+3 y=-8 $$.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}2x+5y=1\2x-3y=4\2x+3y=-8\end{array}\right.$$
- step1: Solve the equation:
  $$\left\{ \begin{array}{l}x=\frac{1-5y}{2}\2x-3y=4\2x+3y=-8\end{array}\right.$$
- step2: Substitute the value of $$x:$$
  $$\left\{ \begin{array}{l}2\times \frac{1-5y}{2}-3y=4\2\times \frac{1-5y}{2}+3y=-8\end{array}\right.$$
- step3: Simplify:
  $$\left\{ \begin{array}{l}1-8y=4\1-2y=-8\end{array}\right.$$
- step4: Solve the equation:
  $$\left\{ \begin{array}{l}y=-\frac{3}{8}\1-2y=-8\end{array}\right.$$
- step5: Substitute the value of $$y:$$
  $$1-2\left(-\frac{3}{8}\right)=-8$$
- step6: Simplify:
  $$\frac{7}{4}=-8$$
- step7: Calculate:
  $$\left\{ \begin{array}{l}x \in \varnothing \y \in \varnothing \end{array}\right.$$
- step8: Rewrite:
  $$(x, y) \in \varnothing$$
Here are the points of intersection for the given pairs of lines:

4. For the equations:
   $$
   \begin{align*}
   x + y &= 5 \
   2x - y &= 1
   \end{align*}
   $$
   The point of intersection is $$ (x, y) = (2, 3) $$.

5. For the equations:
   $$
   \begin{align*}
   x - y &= 2 \
   2x + y &= 7
   \end{align*}
   $$
   The point of intersection is $$ (x, y) = (3, 1) $$.

6. For the equations:
   $$
   \begin{align*}
   2x + 5y &= 1 \
   2x - 3y &= 4 \
   2x + 3y &= -8
   \end{align*}
   $$
   There is no solution (the lines are parallel).

7. For the equations:
   $$
   \begin{align*}
   3x - 2y &= 17 \
   4x + 3y &= 0 \
   x + 3y &= 13 \
   2x + 5y &= 21
   \end{align*}
   $$
   There is no solution (the lines are parallel).

In summary:
- The intersection points are $$ (2, 3) $$ and $$ (3, 1) $$.
- The other pairs of equations do not intersect (no solution).
The points of intersection are: - $$ (2, 3) $$ - $$ (3, 1) $$ The other pairs of equations do not intersect.

Use simultaneous equations to find the point of intersection of the following pairs of lines:
4.

5.

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Use simultaneous equations to find the point of intersection of the following pairs of lines: 4. 5.