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\( \begin{array}{ll}-3-f=5 & \text { 8. } 9-v=0 \\ \text { 9. } \\ -4-e=-1 & \text { 10. } b+9=8 \\ \text { 11. } h+-5=0 & \text { 12. }-6-p=-1 \\ \text { 13. } 8-x=-1 & \text { 14. }-2-j=5 \\ \text { 15. }-3-y=-2 & \text { 16. } z+1=0 \\ \text { 17. }-4-m=-1 & \text { 18. } t+0=-1 \\ \text { 19. } 5-q=4 & \text { 20. } o+-4=-\end{array} \)

Ask by Chandler Hills. in New Zealand
Mar 20,2025

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Solutions for the equations are: - \( f = -8 \) - \( v = 9 \) - \( e = -3 \) - \( b = -1 \) - \( h = 5 \) - \( p = -5 \) - \( x = 9 \) - \( j = -7 \) - \( y = -1 \) - \( z = -1 \) - \( m = -3 \) - \( t = -1 \) - \( q = 1 \) The 20th equation is incomplete and cannot be solved.

Solución

1. Solve \(-3-f=5\):   Add \(3\) to both sides:   \(\displaystyle -f = 5+3=8\)   Multiply by \(-1\):   \(\displaystyle f=-8\) 2. Solve \(9-v=0\):   Subtract \(9\) from both sides:   \(\displaystyle -v=-9\)   Multiply by \(-1\):   \(\displaystyle v=9\) 3. Solve \(-4-e=-1\):   Add \(4\) to both sides:   \(\displaystyle -e=-1+4=3\)   Multiply by \(-1\):   \(\displaystyle e=-3\) 4. Solve \(b+9=8\):   Subtract \(9\) from both sides:   \(\displaystyle b=8-9=-1\) 5. Solve \(h+-5=0\) (i.e. \(h-5=0\)):   Add \(5\) to both sides:   \(\displaystyle h=5\) 6. Solve \(-6-p=-1\):   Add \(6\) to both sides:   \(\displaystyle -p=-1+6=5\)   Multiply by \(-1\):   \(\displaystyle p=-5\) 7. Solve \(8-x=-1\):   Subtract \(8\) from both sides:   \(\displaystyle -x=-1-8=-9\)   Multiply by \(-1\):   \(\displaystyle x=9\) 8. Solve \(-2-j=5\):   Add \(2\) to both sides:   \(\displaystyle -j=5+2=7\)   Multiply by \(-1\):   \(\displaystyle j=-7\) 9. Solve \(-3-y=-2\):   Add \(3\) to both sides:   \(\displaystyle -y=-2+3=1\)   Multiply by \(-1\):   \(\displaystyle y=-1\) 10. Solve \(z+1=0\):   Subtract \(1\) from both sides:   \(\displaystyle z=-1\) 11. Solve \(-4-m=-1\):   Add \(4\) to both sides:   \(\displaystyle -m=-1+4=3\)   Multiply by \(-1\):   \(\displaystyle m=-3\) 12. Solve \(t+0=-1\) (i.e. \(t=-1\)):   Thus, \(\displaystyle t=-1\) 13. Solve \(5-q=4\):   Subtract \(5\) from both sides:   \(\displaystyle -q=4-5=-1\)   Multiply by \(-1\):   \(\displaystyle q=1\) 14. The 20th equation is given as \(o+-4=-\) which is incomplete and does not provide enough information to solve for \(o\). Thus, the solutions are: \[ \begin{aligned} f &= -8,\\ v &= 9,\\ e &= -3,\\ b &= -1,\\ h &= 5,\\ p &= -5,\\ x &= 9,\\ j &= -7,\\ y &= -1,\\ z &= -1,\\ m &= -3,\\ t &= -1,\\ q &= 1, \end{aligned} \] and the equation for \(o\) is incomplete.

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Extra Insights

To solve these equations, let's start with a little background! Equations are like puzzles where you need to find the value of unknowns, usually represented by letters. These basic algebraic operations have been around for centuries and were formalized by mathematicians like Al-Khwarizmi in the 9th century. It’s like a treasure hunt, where each step gets you closer to finding the hidden treasure, which in this case is the value of the variable. Now, once you’ve cracked these hidden values, how do they show up in the real world? Whether you're budgeting your monthly expenses or measuring ingredients in a recipe, algebra equations come into play! For instance, if you know the cost of two apples is $3, you can use equations to find out how much one apple costs by setting up and solving the equation \(2x = 3\). It's a universal language that helps us decipher the many problems life throws at us!

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