One way to solve this is to know that the two equations of a linear system without a solution - an inconsistent system - are represented by a pair of parallel lines, which are lines with the same slope. To find the slope of the line of the equation \(\displaystyle 4x+ 9y = 21\), rewrite the equation in the slope-intercept form \(\displaystyle y = mx+b\), with \(\displaystyle m\) the slope of the line. This is done by solving for \(\displaystyle y\), so isolate \(\displaystyle y\) in the equation as follows:

\(\displaystyle 4x+ 9y = 21\)

\(\displaystyle 4x+ 9y - 4x = 21 - 4x\)

\(\displaystyle 9y = - 4x + 21\)

\(\displaystyle \frac{9y }{9}= \frac{- 4x + 21}{9}\)

\(\displaystyle y =- \frac{4 }{9}x+ \frac{ 21}{9}\)

The slope of the line is \(\displaystyle - \frac{4 }{9}\), the coefficient of \(\displaystyle x\).

The equations in all four choices are already in slope-intercept form, so examine the coefficient of \(\displaystyle x\) in each. The equation \(\displaystyle y = -\frac{4}{9}x+ 21\) is the only one with \(\displaystyle x\)-coefficient \(\displaystyle - \frac{4 }{9}\), so its slope is also \(\displaystyle - \frac{4 }{9}\). Since the lines of the equations are of the same slope, they are parallel, and the equations form a system of equations with no solution. The correct response is therefore \(\displaystyle y = -\frac{4}{9}x+ 21\).

There are three variables, but only 2 equations. With fewer equations than variables, it is not possible to solve for the value of each variable separately, but we can eliminate variables to manipulate the equation into the form we want.

Multiply the second equation by \(\displaystyle -2\) on both sides, resulting in \(\displaystyle -2y+2z=-14\).

Then add the equations together (allowed since we are adding the same thing to both sides, just in a different form):

\(\displaystyle (x+2y)+(-2y+2z)=6+(-14))\)

Simplify:

\(\displaystyle x+2z=-8\)

Resist the urge to keep working, since that's exactly what they've asked for. Done!

Use the addition method to eliminate one variable and therefore solve the system of equations. 

\(\displaystyle -3x-2y=48\)

\(\displaystyle 3x+7y=-63\)

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\(\displaystyle 5y=-15\)

\(\displaystyle y=-3\)

Now substitute to find x.

\(\displaystyle -3x-2(-3)=48\)

\(\displaystyle -3x+6=48\)

\(\displaystyle -3x=42\)

\(\displaystyle x=-14\)

\(\displaystyle (-14,-3)\)

Translate the word problem into a system of equations:

\(\displaystyle A+O=55\)

\(\displaystyle 0.55A+0.65O=33.45\)

Use the first equation to solve for A in terms of O, or vice versa. 

\(\displaystyle A=55-O\)

Use substitution to solve for O in the second equation.

\(\displaystyle 0.55(55-O)+0.65O=33.45\)

\(\displaystyle 30.25-0.55O+0.65O=33.45\)

\(\displaystyle 30.25+0.1O=33.45\)

\(\displaystyle 0.1O=3.2\)

\(\displaystyle O=32\)

Now plug the value for O back into the first equation to solve for A.

\(\displaystyle A+32=55\)

\(\displaystyle A=23\)

Subtract \(\displaystyle 9x\) and \(\displaystyle 72\) from the both sides to get \(\displaystyle -6x = 48\). 

Divide both sides by \(\displaystyle -6\), to get

 \(\displaystyle x=\frac{48}{-6}=-8\)

As is clear from the graph, in the interval between \(\displaystyle -2\) (\(\displaystyle -2\) included) to \(\displaystyle 1\), the \(\displaystyle f(x)\) is constant at \(\displaystyle 1\) and then from \(\displaystyle x=1\) (\(\displaystyle 1\) not included) to \(\displaystyle x=3\) (\(\displaystyle 3\) not included), the \(\displaystyle f(x)\) is a decreasing function.

To begin, we analyze the equation given: the base equation, \(\displaystyle y=x^{2}\) is shifted left one unit and vertically stretched by a factor of 2. The graph of the equation \(\displaystyle y\geq2(x+1)^2\) is:

To solve the inequality, we need to take a test point and plug it in to see if it matches the inequality. The only points that cannot be used are those directly on our parabola, so let's use the origin \(\displaystyle (0,0)\). If plugging this point in makes the inequality true, then we shade the area containing that point (in this case, outside the parabola); if it makes the inequality untrue, then the opposite side is shaded (in this case, the inside of the parabola). Plugging the numbers in shows:

\(\displaystyle 0\geq2(0+1)^2\)

Simplified as:

\(\displaystyle 0\geq2\)

Which is not true, so the area inside of the parabola should be shaded, resulting in the following graph:

Let x be the number of pencils and y be the number of pens Sally purchased. We are told that each pencil costs 25 cents. Because, the total amount she spent was given in dollars, we want to convert 25 cents to dollars. Because there are 100 cents in one dollar, 25 cents  = $0.25. Similarly, each pen costs $0.50.

The amount that Sally spent on pencils is equal to the product of the number of pencils and the cost per pencil. We can model the amount of money spent on pencils as 0.25x.

Likewise, the amount Sally spent on pens is equal to the product of the number of pens and the cost per pen, or 0.5y.

Since we are told that Sally spent $5.75 total on pens and pencils combined, we can write the following:

\(\displaystyle 0.25x + 0.5y = 5.75\)

We are also told that the number of pens is one greater than the number of pencils. We can thus write:

\(\displaystyle y = x + 1\)

We now have two equations and two unknowns. In order to solve this system of equations, we can take the value of y = x + 1 and substitute it into the other equation.

\(\displaystyle 0.25x + 0.5(x + 1) = 5.75\)

Distribute.

\(\displaystyle 0.25x + 0.5x + 0.5 = 5.75\)

Combine x terms.

\(\displaystyle 0.75x + 0.5 = 5.75\)

Subtract 0.5 from both sides.

\(\displaystyle 0.75x = 5.25\)

Divide both sides by 0.75.

\(\displaystyle \\x = 7 \\y = x + 1 = 8\)

This means Sally bought 7 pencils and 8 pens. The question asks us to find the number of pens and pencils purchased altogether, which would equal

\(\displaystyle 7 + 8=15\)

The answer is 15.

First, solve this equation for y and then substitute the answer into the second equation:

\(\displaystyle \\4x = 2y \\\\y = \frac{4x}{2}=2x\)

Now substitute into the second equation and solve for x:

\(\displaystyle \\y + x = 10 \\2x + x = 10\)

To solve for x, add the coefficients on the x variables together then divide both sides by three. 

\(\displaystyle \\2x+x=10 \\3x=10 \\\\x=\frac{10}{3}\)

Set \(\displaystyle x\) as the number of people and \(\displaystyle y\) as the number of dogs.

The head equation is going to be:

\(\displaystyle x + y = 45\)

The legs equation is going to be:

\(\displaystyle 2x + 4y = 140\) 

Manipulating the first equation to solve for \(\displaystyle x\) results in the following.

\(\displaystyle \\x + y{\color{Red} -y} = 45{\color{Red} -y} \\x=45-y\)

Plugging in

\(\displaystyle x = 45 - y\) into the second equation and solving for \(\displaystyle y\), you get

\(\displaystyle \\2(45-y)+4y=140 \\90-2y+4y=140 \\90+2y=140 \\2y=50 \\y=25\)