Precalculus : Inequalities and Linear Programming

Study concepts, example questions & explanations for Precalculus

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Example Questions

Example Question #1 : Systems Of Equations

Solve the following system of linear equations:

\(\displaystyle 3x+2y=6\)

\(\displaystyle x-y=-1\)

Possible Answers:

\(\displaystyle \left(-\frac{2}{7},\frac{3}{7}\right)\)

\(\displaystyle \left(\frac{4}{5},\frac{9}{5}\right)\)

\(\displaystyle \left(\frac{1}{3},\frac{2}{3}\right)\)

\(\displaystyle (4,3)\)

\(\displaystyle \left(\frac{3}{2},-\frac{5}{2}\right)\)

Correct answer:

\(\displaystyle \left(\frac{4}{5},\frac{9}{5}\right)\)

Explanation:

In order to solve a system of linear equations, we must start by solving one of the equations for a single variable:

\(\displaystyle x-y=-1\rightarrow y=x+1\)

We can now substitute this value for y into the other equation and solve for x:

\(\displaystyle 3x+2y=6\rightarrow 3x+2(x+1)=6\rightarrow 5x=4\rightarrow x=\frac{4}{5}\)

Our last step is to plug this value of x into either equation to find y:

\(\displaystyle y=x+1=\frac{4}{5}+1=\frac{9}{5}\)

Example Question #1 : Systems Of Equations

Solve the system of linear equations for \(\displaystyle x\):

\(\displaystyle y=2x+1\)

\(\displaystyle y=x-9\)

Possible Answers:

\(\displaystyle x=1\)

\(\displaystyle x=10\)

\(\displaystyle x=-5\)

\(\displaystyle x=-10\)

\(\displaystyle x=5\)

Correct answer:

\(\displaystyle x=-10\)

Explanation:

We first move \(\displaystyle x's\) to the left side of the equation:

\(\displaystyle y-2x=1\)

\(\displaystyle y-x=-9\)

 

Subtract the bottom equation from the top one:

Left Side:

\(\displaystyle y-2x-(y-x)=y-2x-y+x=-x\)

 

Right Side:

\(\displaystyle 1-(-9)=10\)

 

So

\(\displaystyle -x=10\)

 

So dividing by a -1 we get our result.

\(\displaystyle x=-10\)

Example Question #2 : Systems Of Equations

Solve the following system of linear equations:

\(\displaystyle y=3x-2\)

\(\displaystyle y=-x-6\)

Possible Answers:

\(\displaystyle (3,5)\)

\(\displaystyle (5,1)\)

\(\displaystyle (-4,6)\)

\(\displaystyle (-6,4)\)

\(\displaystyle (-1,-5)\)

Correct answer:

\(\displaystyle (-1,-5)\)

Explanation:

For any system of linear equations, we can start by solving one equation for one of the variables, and then plug its value into the other equation. In this system, however, we can see that both equations are equal to y, so we can set them equal to each other:

\(\displaystyle y=3x-2\)

\(\displaystyle y=-x-6\)

\(\displaystyle 3x-2=-x-6\rightarrow 4x=-4\rightarrow x=-1\)

Now we can plug this value for x back into either equation to solve for y:

\(\displaystyle y=3x-2=3(-1)-2=-5\)

So the solutions to the system, where the lines intersect, is at the following point:

\(\displaystyle (-1.-5)\)

Example Question #1 : Systems Of Equations

Solve the following system of linear equations:

\(\displaystyle 7x-y=-2\)

\(\displaystyle 3=\frac{3}{2}y+6x\)

Possible Answers:

\(\displaystyle (3,6)\)

\(\displaystyle (4,2)\)

\(\displaystyle (0,2)\)

\(\displaystyle (5,3)\)

\(\displaystyle (7,0)\)

Correct answer:

\(\displaystyle (0,2)\)

Explanation:

In order to solve a system of linear equations, we can either solve one equation for one of the variables, and then substitute its value into the other equation, or we can solve both equations for the same variable so that we can set them equal to each other. Let's solve both equations for y so that we can set them equal to each other:

\(\displaystyle 7x-y=-2\rightarrow y=7x+2\)

\(\displaystyle 3=\frac{3}{2}y+6x\rightarrow \frac{3}{2}y=-6x+3\rightarrow y=-4x+2\)

\(\displaystyle 7x+2=-4x+2\rightarrow 11x=0\rightarrow x=0\)

Now we just plug our value for x back into either equation to find y:

\(\displaystyle y=7x+2=7(0)+2=2\)

So the solution to the system is the point:

\(\displaystyle (0,2)\)

Example Question #1 : Inequalities And Linear Programming

Use back substitution to solve the system of linear equations.

 

\(\displaystyle \newline 2x+3y+z=17 \newline 2y-3z=20 \newline z=-6\)

 

Possible Answers:

\(\displaystyle x=10, y=1, z=-6\)

\(\displaystyle x=2,y=-1,z=-6\)

\(\displaystyle x=8, y=0, z=-6\)

\(\displaystyle x=8,y=-1,z=1\)

Correct answer:

\(\displaystyle x=10, y=1, z=-6\)

Explanation:

Start from equation 3 because it has the least number of variables. We see directly that \(\displaystyle z=-6\).

Back substitute into the equation with the next fewest variables, equation 2. Then,

\(\displaystyle 2y-3(-6)=20\). Solving for \(\displaystyle y\), we get 

\(\displaystyle 2y+18=20\) or \(\displaystyle y=1\).

Then back substitute our \(\displaystyle y\) and \(\displaystyle z\) into equation 1 to get

\(\displaystyle 2x+3(1)+(-6)=17\).

Solving for x,

\(\displaystyle x=10\).

So our solution to the system is

\(\displaystyle x=10, y=1, z=-6\)

Example Question #1 : Systems Of Equations

Solve the following system:

\(\displaystyle x+y=12\)

\(\displaystyle x+3y=4\)

Possible Answers:

\(\displaystyle (-4,16)\)

\(\displaystyle (16,-4)\)

\(\displaystyle (8,-4)\)

\(\displaystyle (16,4)\)

Correct answer:

\(\displaystyle (16,-4)\)

Explanation:

We can solve the system using elimination. We can eliminate our \(\displaystyle x\) by multiplying the top equation by \(\displaystyle -1\):

\(\displaystyle -x-y=-12\)

 

and then adding it to the bottom equation:

 \(\displaystyle -x-y=-12\)

    \(\displaystyle x+3y=4\)

____________________

           \(\displaystyle 2y=-8\)

            \(\displaystyle y=-4\)

We can now plug in our y-value into the top equation and solve for our x-value:

\(\displaystyle x+y=12\)

\(\displaystyle x-4=12\)

\(\displaystyle x=16\)

 

Our solution is then \(\displaystyle (16,-4)\)

Example Question #1 : Systems Of Equations

Solve the following system:

\(\displaystyle 7x+9y=38\)

\(\displaystyle x=4-y\)

Possible Answers:

\(\displaystyle (-1,-5)\)

\(\displaystyle (1,5)\)

\(\displaystyle (-1,5)\)

\(\displaystyle (5,-1)\)

Correct answer:

\(\displaystyle (-1,5)\)

Explanation:

We can solve the system using substitution since the bottom equation is already solved for \(\displaystyle x\):

\(\displaystyle 7(4-y)+9y=38\)

\(\displaystyle 28-7y+9y=38\)

\(\displaystyle 28+2y=38\)

\(\displaystyle 2y=10\)

\(\displaystyle y=5\)

 

Now we can plug in our value into the bottom equation to find our x-value:

\(\displaystyle x=4-y=4-5=-1\)

 

So our solution is \(\displaystyle (-1,5)\)

Example Question #3 : Solve Systems Of Linear Equations

Solve the following system:

\(\displaystyle 2x-5y=7\)

\(\displaystyle x-3y=-1\)

Possible Answers:

\(\displaystyle (-9,26)\)

\(\displaystyle (26,9)\)

\(\displaystyle (9,26)\)\(\displaystyle (9,26)\)

\(\displaystyle (-26,9)\)

Correct answer:

\(\displaystyle (26,9)\)

Explanation:

We can solve the system using elimination. We can eliminate the x terms by multiplying the bottom equation by \(\displaystyle -2\):

\(\displaystyle -2x+6y=2\)

 

and now add it to the top equation:

   \(\displaystyle 2x-5y=7\)

\(\displaystyle -2x+6y=2\)

__________________

               \(\displaystyle y=9\)

 

We plug in our y-value into the bottom equation to get our x-value:

\(\displaystyle x-3(9)=-1\)

\(\displaystyle x-27=-1\)

\(\displaystyle x=26\)

 

Our solution is then \(\displaystyle (26,9)\)

Example Question #3 : Solve Systems Of Linear Equations

Solve the following system:

\(\displaystyle 2y=5x-7\)

\(\displaystyle 4x=2y-10\)

Possible Answers:

\(\displaystyle (-17,39)\)

\(\displaystyle (17,-39)\)

\(\displaystyle (17,39)\)

\(\displaystyle (39,17)\)

Correct answer:

\(\displaystyle (17,39)\)

Explanation:

We can solve this system using either substitution or elimination. We'll eliminate them here.

Note: If you wanted to do substitution, we can do it by substituting the top equation into the bottom for \(\displaystyle 2y\).

 

We'll rearrange the bottom equation to have both y-values aligned and then add the equations:

   \(\displaystyle 2y=5x-7\)

\(\displaystyle -2y=-4x-10\)

_____________________

      \(\displaystyle 0=x-17\)

      \(\displaystyle x=17\)

 

Now that we have our x-value, we can find our y-value:

\(\displaystyle 2y=5(17)-7\)

\(\displaystyle 2y=85-7\)

\(\displaystyle y=39\)

Our answer is then \(\displaystyle (17,39)\)

Example Question #10 : Inequalities And Linear Programming

Solve the following system of equations:

\(\displaystyle x+y=4\)

\(\displaystyle x-y=6\)

Possible Answers:

\(\displaystyle x=3,y=1\)

\(\displaystyle x=5,y=-1\)

\(\displaystyle x=1,y=3\)

\(\displaystyle x=-1,y=5\)

Correct answer:

\(\displaystyle x=5,y=-1\)

Explanation:

There are many ways to solve this system of equations. The following is just one way to reach the answer.

Add the two together, to elimnate the y variable. Solve for x and then plug it back in to the first equation to solve for y.

\(\displaystyle x+y=4\)

\(\displaystyle x-y=6\)

 

\(\displaystyle 2x=10\Rightarrow x=5\)

\(\displaystyle 5+y=4\Rightarrow y=-1\)

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