Algebra I

Slope is defined as the steepness of a line and can be calculated using the formula “rise over run.” More precisely, this formula is equal to the change in $y$-value (“rise”) divided by the change in $x$-value (“run”) between two points. To find these changes in values, use $y_2-y_1$ and $x_2-x_1$, where $(x_1,y_1)$ and $(x_2,y_2)$ are points on the line.

$m=\frac{\text{rise}}{\text{run}}=\frac{y_2-y_1}{x_2-x_1}$

In this case, let $(x_1,y_1)=(0,-3)$ and $(x_2,y_2)=(4,5)$. Plug each value into the slope formula and simplify.

$m=\frac{5-(-3)}{4-0}=\frac{5+3}{4-0}=\frac{8}{4}=2$

Therefore, the slope of this line is 2.

Equations written in slope-intercept form are in the format $y=mx+b$, where $m$ is the slope and $b$ is the $y$-intercept. In this situation, the cost per square foot is $100. Therefore, the slope $m$ is equal to 100. This can already be seen in the point-slope equation.

$y-y_1=\mathbf{m}(x-x_1)$

$y-\mathrm{205,000}=\mathbf{100}(x-\mathrm{2,000})$

The quickest way to determine $b$’s value is to rearrange the point-slope equation so that $y$ is by itself on the left and like terms are combined on the right. Add 205,000 to both sides of the equation.

$y=100(x-\mathrm{2,000})+\mathrm{205,000}$

Now, distribute the 100 to the $x$ and the –2,000.

$y=100x-\mathrm{200,000}+\mathrm{205,000}$

Finally, combine like terms to see that $b=\mathrm{5,000}$.

$y=100x+\mathrm{5,000}$

This equation is now in slope-intercept form. From it we can quickly determine that the price of each property is equal to the square-footage times $100, plus a fixed cost of $5,000.

To simplify, combine like terms with $x$’s and $y$’s and the same exponent. In those problems, you can only combine terms that have the exact same variable.

This expression has several terms with the variable $x$, although to varying powers. Look for terms that have the same power of $x$ and group them.

The highest power in this expression is $x^3$, and there are two terms which have it: $2x^3$ and $x^3$. These may be combined to obtain $3x^3$.

$6x+\mathbf{2}{\mathbf{x}}^{\mathbf{3}}-5x^2-3x^2+{\mathbf{x}}^{\mathbf{3}}-5x$

$3x^3\dots$

The next power to look for is $x^2$, which is seen in the terms $-5x^2$ and $-3x^2$. These terms combine to form $-8x^2$.

$6x+2x^3-\mathbf{5}{\mathbf{x}}^{\mathbf{2}}-\mathbf{3}{\mathbf{x}}^{\mathbf{2}}+x^3-5x$

$3x^3-8x^2\dots$

Finally, the remaining terms both have $x$ to the first power. 6x and -5x combine to make $x$.

$\mathbf{6}\mathbf{x}+2x^3-5x^2-3x^2+x^3-\mathbf{5}\mathbf{x}$

$3x^3-8x^2+x$

To get $x$ by itself, first move –4 to the other side of the equation by adding 4 to both sides.

$\sqrt{x+10}=2+4$

$\sqrt{x+10}=6$

The next step is getting rid of the square root symbol. A square root can be undone by squaring it. Square both sides of the equation.

${\left(\sqrt{x+10}\right)}^2={\left(6\right)}^2$

$x+10=36$

Subtract 10 from both sides.

$x=26$

When working with roots, it is always important to check for extraneous solutions. Do this by substituting $x=26$ into the original equation and checking to see if the statement is true or not.

$\sqrt{\left(26\right)+10}-4=2$

$\sqrt{36}-4=2$

$6-4=2$

$2=2$

Since $2=2$ is a correct statement, $x=26$ is a solution to the equation.

The percent change from one observation to the next can be found using the formula $\text{percent change}=\frac{\text{new}-\text{old}}{\text{old}}\times 100\mathrm{\%}$. The numerator, $\text{new}-\text{old}$, represents the change, or difference, in the observations. A positive amount means that there is an increase, while a negative amount indicates a decrease. In this scenario, the new amount is 92 and the old amount is 80. From this it can be seen that there was an increase of 12 sno-cones.

$\frac{\text{new}-\text{old}}{\text{old}}\times 100\mathrm{\%}$

$=\frac{92-80}{\text{old}}\times 100\mathrm{\%}$

$=\frac{12}{\text{old}}\times 100\mathrm{\%}$

This amount is divided by the old amount, 80, to determine the ratio of the change to the original amount. Here, $\frac{12}{80}$ reduces to $\frac{3}{20}$, or equivalently 0.15.

$=\frac{12}{\text{old}}\times 100\mathrm{\%}$

$=\frac{12}{80}\times 100\mathrm{\%}$

$=0.15\times 100\mathrm{\%}$

The final step in using the percent change formula is multiplying by 100 to convert from a decimal to a percentage. Here, $0.15\times 100=15\mathrm{\%}$.

$0.15\times 100\mathrm{\%}=15$

The expression $8x-352$ represents net income, and the inequality that sets it greater than or equal to zero restricts the value of $x$ (candles sold) so that net income is not negative. To determine at least how many candles are required to break even, solve for $x$. This process is nearly identical to solving for $x$ when there is an equals sign rather than an inequality sign.

First, add 352 to both sides.

$8x-352+352\ge 0+352$

$8x\ge 352$

Now, divide both sides by 8 to get $x$ by itself.

$\frac{8x}{8}\ge \frac{352}{8}$

$x\ge 44$

This means that Sheila’s Candle Co. has a net profit of zero dollars or greater when 44 candles are sold.

When graphing linear inequalities, a helpful first step is drawing the line on the graph. In this case, the line is $y=-2x+1$. It has a slope of –2 and a $y$-intercept at 1. To plot this, mark the $y$-intercept at the point $(0,1)$, and then use the slope, $\frac{-2}{1}$, to mark another point that is down two spaces and to the right one space. This will go to the point $(1,-1)$.

Notice that the inequality $y>-2x+1$ has a greater-than sign rather than a greater-than-or-equal-to sign. Because of this, the line drawn on the graph will be a dashed line. Use a dashed line to connect the points $(0,1)$ and $(1,-1)$.

The final step in plotting this inequality is denoting whether the appropriate area lies above or below this dashed line. A quick way to make this determination is by noting that since the greater-than symbol $(>)$ is used, the area is greater than, or above, the line.

This step may be checked by plugging the point $(0,0)$ into the inequality and seeing whether it is included in the shaded area.

$y>-2x+1$
$0>-2\left(0\right)+1$
$0>0+1$
$0>1$

Zero is not larger than 1. This informs us that the point $(0,0)$ is not included in the shaded region. Since $(0,0)$ is below the dashed line, we would instead shade the area that is above the dashed line.

To add together $A\left(x\right)$ and $B\left(x\right)$, write out their equations with a plus sign between them.

$A\left(x\right)+B\left(x\right)=(-x^2+400x-\mathrm{20,000})+(-x^2+600x-\mathrm{65,000})$

Since this situation deals with addition rather than subtraction, the parentheses may be dropped.

$-x^2+400x-\mathrm{20,000}-x^2+600x-\mathrm{65,000}$

The next step is to combine like terms based on their powers of $x$. Combine like terms to find the profit function for Models A and B.

$-2x^2+\mathrm{1,000}x-\mathrm{85,000}$

Each of the four functions shown are written in vertex form.

$f\left(x\right)=a{(x-h)}^2+k$

In vertex form, h and k are the x- and y-coordinates of the vertex, respectively. That is, $(h,k)$ is the vertex. The variable a denotes a coefficient.

Because we want a vertex at the point $(-1,3)$, let $h=-1$ and $k=3$. Plug these values into vertex form.

$f\left(x\right)=a{(x-(-1))}^2+3$

Now, simplify.

$f\left(x\right)=a{(x+1)}^2+3$

Since a is a coefficient, it can be any number. Out of the four options given, only one follows this structure, choice B: $f\left(x\right)=-2{(x+1)}^2+3$.

Choice A is not correct because its vertex is at the point $(1,3)$. Choice C is not correct because its vertex is at $(1,0)$. Choice D is not correct because its vertex is at $(3,-1)$.

Systems of equations can be solved using either substitution or elimination. Since these equations do not have a common $x$-coefficient or $y$-coeffient, it will likely be quicker to use substitution.

The first equation can be solved for $x$ by moving the $y$-term to the other side.

$x-10y=-24$

$x=-24+10y$

Now that we know that $x=-24+10y$, plug this value into the second equation in place of $x$.

$4x-2y=-1$

$4(-24+10y)-2y=-1$

Simplify by distributing and combining like terms.

$-96+40y-2y=-1$

$-96+38y=-1$

Solve for $y$.

$38y=95$

$y=\frac{95}{38}=2.5$

We now know that the solution to the system of equations has a $y$-value of 2.5. Use this information to determine the $x$-value of the solution. Since $x=-24+10y$ and $y=2.5$, substitute 2.5 for $y$ and solve for $x$.

$x=-24+10\left(2.5\right)$

$x=-24+25$

$x=1$

This indicates that after one day, the bamboo catches up to the tomato plant at a height of 2.5 feet. The system of equations can also be graphed using two lines, as shown below. The point at which the two lines intersect is the solution to the system.