2.2 Linear Equations in One Variable

Solve linear equations using a general strategy.

$2 \left(w + 5\right) + 1 = 10 + 4 w + 2$

Solve the following equation: $2 x + 7 = 19.$

Solve the linear equation in one variable: $2 x + 1 = −9.$

Solve the following equation: $4 \left(x −3\right) + 12 = 15 −5 \left(x + 6\right) .$

Solve the equation in one variable: $−2 \left(3 x - 1\right) + x = 14 - x .$

Solve the rational equation: $\frac{7}{2 x} - \frac{5}{3 x} = \frac{22}{3} .$

Solve the following rational equation:

Solve the rational equation: $\frac{2}{3 x} = \frac{1}{4} - \frac{1}{6 x} .$

Solve the following rational equation: $\frac{1}{x} = \frac{1}{10} - \frac{3}{4 x} .$

Solve the rational equation: $- \frac{5}{2 x} + \frac{3}{4 x} = - \frac{7}{4} .$

Solve the following rational equations and state the excluded values:

1. ⓐ $\frac{3}{x - 6} = \frac{5}{x}$
2. ⓑ $\frac{x}{x - 3} = \frac{5}{x - 3} - \frac{1}{2}$
3. ⓒ $\frac{x}{x - 2} = \frac{5}{x - 2} - \frac{1}{2}$

Solve $\frac{- 3}{2 x + 1} = \frac{4}{3 x + 1} .$ State the excluded values.

Solve the rational equation after factoring the denominators: $\frac{2}{x + 1} - \frac{1}{x - 1} = \frac{2 x}{x^{2} - 1} .$ State the excluded values.

Solve the rational equation: $\frac{2}{x - 2} + \frac{1}{x + 1} = \frac{1}{x^{2} - x - 2} .$

Find the slope of a line that passes through the points $\left(2 , −1\right)$ and $\left(−5 , 3\right) .$

Find the slope of the line that passes through the points $\left(−2 , 6\right)$ and $\left(1 , 4\right) .$

Identify the slope and y-intercept, given the equation $y = - \frac{3}{4} x - 4.$

Write the equation of the line with slope $m = −3$ and passing through the point $\left(4 , 8\right) .$ Write the final equation in slope-intercept form.

Given $m = 4 ,$ find the equation of the line in slope-intercept form passing through the point $\left(2 , 5\right) .$

Find the equation of the line passing through the points $\left(3 , 4\right)$ and $\left(0 , −3\right) .$ Write the final equation in slope-intercept form.

Find the equation of the line with $m = −6$ and passing through the point $\left(\frac{1}{4} , −2\right) .$ Write the equation in standard form.

Find the equation of the line in standard form with slope $m = - \frac{1}{3}$ and passing through the point $\left(1 , \frac{1}{3}\right) .$

Find the equation of the line passing through the given points: $\left(1 , −3\right)$ and $\left(1 , 4\right) .$

Find the equation of the line passing through $\left(−5 , 2\right)$ and $\left(2 , 2\right) .$

Find the equation of the line parallel to $5 x = 7 + y$ and passing through the point $\left(−1 , −2\right) .$

Find the equation of the line perpendicular to $5 x - 3 y + 4 = 0$ and passing through the point $\left(- 4 , 1\right) .$

What does it mean when we say that two lines are parallel?

How do we recognize when an equation, for example $y = 4 x + 3 ,$ will be a straight line (linear) when graphed?

When solving the following equation:

$\frac{2}{x - 5} = \frac{4}{x + 1}$

explain why we must exclude $x = 5$ and $x = −1$ as possible solutions from the solution set.

Solve the equation for $x .$

$4 x - 3 = 5$

Solve the equation for $x .$

$12 - 5 \left(x + 3\right) = 2 x - 5$

Solve the equation for $x .$

$\frac{x}{3} - \frac{3}{4} = \frac{2 x + 3}{12}$

Solve the equation for $x .$

$3 \left(2 x - 1\right) + x = 5 x + 3$

Solve the equation for $x .$

$\frac{x + 2}{4} - \frac{x - 1}{3} = 2$

Solve each rational equation for $x .$ State all x-values that are excluded from the solution set.

$2 - \frac{3}{x + 4} = \frac{x + 2}{x + 4}$

Solve each rational equation for $x .$ State all x-values that are excluded from the solution set.

$\frac{3 x}{x - 1} + 2 = \frac{3}{x - 1}$

Solve each rational equation for $x .$ State all x-values that are excluded from the solution set.

$\frac{1}{x} = \frac{1}{5} + \frac{3}{2 x}$

Find the equation of the line using the point-slope formula. Write all the final equations using the slope-intercept form.

$\left(1 , 2\right)$ with a slope of $- \frac{4}{5}$

Find the equation of the line using the point-slope formula. Write all the final equations using the slope-intercept form.

y-intercept is 2, and $\left(4 , −1\right)$

Find the equation of the line using the point-slope formula. Write all the final equations using the slope-intercept form.

$\left(1 , 3\right) \textrm{ }\text{and}\textrm{ }\textrm{ } \left(5 , 5\right)$

Find the equation of the line using the point-slope formula. Write all the final equations using the slope-intercept form.

perpendicular to $\text{3} y = x - 4$ and passes through the point $\left(−2 , 1\right)$ .

Find the equation of the line using the given information.

$\left(1 , 7\right)$ and $\left(3 , 7\right)$

Find the equation of the line using the given information.

The slope equals zero and it passes through the point $\left(1 , −4\right) .$

Find the equation of the line using the given information.

$\left(–1 , 3\right)$ and $\left(4 , –5\right)$

Find the slope of the line that passes through the given points.

$\left(−3 , 2\right)$ and $\left(4 , −7\right)$

Find the slope of the line that passes through the given points.

$\left(−1 , −2\right)$ and $\left(3 , 4\right)$

Find the slope of the lines that pass through each pair of points and determine whether the lines are parallel or perpendicular.

$\begin{matrix}\left(−1 , 3\right) \textrm{ }\text{and}\textrm{ }\textrm{ } \left(5 , 1\right) \\ \left(−2 , 3\right) \textrm{ }\text{and}\textrm{ }\textrm{ } \left(0 , 9\right)\end{matrix}$

Express the equations in slope intercept form (rounding each number to the thousandths place). Enter this into a graphing calculator as Y1, then adjust the ymin and ymax values for your window to include where the y-intercept occurs. State your ymin and ymax values.

$0.537 x - 2.19 y = 100$

Express the equations in slope intercept form (rounding each number to the thousandths place). Enter this into a graphing calculator as Y1, then adjust the ymin and ymax values for your window to include where the y-intercept occurs. State your ymin and ymax values.

$\frac{200 - 30 y}{x} = 70$

Starting with the standard form of an equation $A x + B y = C$ solve this expression for $y$ in terms of $A , B , C$ and $x$. Then put the expression in slope-intercept form.

Given that the following coordinates are the vertices of a rectangle, prove that this truly is a rectangle by showing the slopes of the sides that meet are perpendicular.

$\left(– 1 , 1\right) , \left(2 , 0\right) , \left(3 , 3\right)$ and $\left(0 , 4\right)$

Use this scenario: The cost of renting a car is $45/wk plus $0.25/mi traveled during that week. An equation to represent the cost would be $y = 45 + .25 x ,$ where $x$ is the number of miles traveled.

What is your cost if you travel 50 mi?

Use this scenario: The cost of renting a car is $45/wk plus $0.25/mi traveled during that week. An equation to represent the cost would be $y = 45 + .25 x ,$ where $x$ is the number of miles traveled.

Suppose you have a maximum of $100 to spend for the car rental. What would be the maximum number of miles you could travel?