Objective 4 - Solve Linear Equations

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Solve linear equations.

Link to section in online textbook and link to section in Prelude to Active Calculus textbook.

Now, watch the video below to review how to solve linear equations. You can use the notes here to follow along with the video and record your thoughts. These techniques will be used throughout most of the semester.

Now try to solve the following linear equations.

Solve the equation below.

$\sage {blocks1[0]} (\sage {displayEquationLeft1}) = \sage {blocks1[3]} (\sage {displayEquationRight1})$

$x = \answer [tolerance=0.01]{\sage {answer1}}$

Did you get $\sage {feedback1}$ as your answer? If so, you are not distributing correctly. Remember that if you have something like $-3(2x-1)$ , you need to distribute the $-3$ to BOTH $2x$ and $-1$ to get $-6x+3$ .

Solve the equation below.

$\sage {blocks2[0]} (\sage {displayEquationLeft2}) = \sage {blocks2[3]} (\sage {displayEquationRight2})$

$x = \answer [tolerance=0.01]{\sage {answer2}}$

Did you get $\sage {feedback2}$ as your answer? If so, you are not distributing correctly. Remember that if you have something like $-3(2x-1)$ , you need to distribute the $-3$ to BOTH $2x$ and $-1$ to get $-6x+3$ .

Solve the equation below.

$\sage {blocks3[0]} (\sage {displayEquationLeft3}) = \sage {blocks3[3]} (\sage {displayEquationRight3})$

$x = \answer [tolerance=0.01]{\sage {answer3}}$

Did you get $\sage {feedback3}$ as your answer? If so, you are not distributing correctly. Remember that if you have something like $-3(2x-1)$ , you need to distribute the $-3$ to BOTH $2x$ and $-1$ to get $-6x+3$ .

Solve the equation below.

$\frac {\sage {displayNumeratorA4}}{\sage {constants4[6]}} - \frac {\sage {displayNumeratorB4}}{\sage {constants4[7]}} = \frac {\sage {displayNumeratorC4}}{\sage {constants4[8]}}$

Adding/Multiplying fractions can be difficult and tedious. Is there something we can multiply both sides of the equation by to remove the fractions from the equation?

$x = \answer [tolerance=0.01]{\sage {answer4}}$

There are two common issues when solving linear equations:

The first we saw in the previous set of problems: not distributing correctly. If you got $\sage {feedback41}$ , check that you distributed any negatives correctly.

The second common issue is not dividing correctly. If we have a fraction like $\frac {6x-4}{2}$ , the 2 is dividing both parts. So, this would become $3x-2$ and not $3x-4$ . If you got $\sage {feedback42}$ , you made this type of mistake.

If you made both of these mistakes, you got $\sage {feedback43}$ . Restart the problem and correct both issues.

Solve the equation below.

$\frac {\sage {displayNumeratorA5}}{\sage {constants5[6]}} - \frac {\sage {displayNumeratorB5}}{\sage {constants5[7]}} = \frac {\sage {displayNumeratorC5}}{\sage {constants5[8]}}$

$x = \answer [tolerance=0.01]{\sage {answer5}}$

There are two common issues when solving linear equations:

The first we saw in the previous set of problems: not distributing correctly. If you got $\sage {feedback51}$ , check that you distributed any negatives correctly.

If you made both of these mistakes, you got $\sage {feedback53}$ . Restart the problem and correct both issues.

Solve the equation below.

$\frac {\sage {displayNumeratorA6}}{\sage {constants6[6]}} - \frac {\sage {displayNumeratorB6}}{\sage {constants6[7]}} = \frac {\sage {displayNumeratorC6}}{\sage {constants6[8]}}$

$x= \answer [tolerance=0.01]{\sage {answer6}}$

There are two common issues when solving linear equations:

The first we saw in the previous set of problems: not distributing correctly. If you got $\sage {feedback61}$ , check that you distributed any negatives correctly.

If you made both of these mistakes, you got $\sage {feedback63}$ . Restart the problem and correct both issues.

Press...	...to do
left/right arrows	Move cursor
shift+left/right arrows	Select region
ctrl+a	Select all
ctrl+x/c/v	Cut/copy/paste
ctrl+z/y	Undo/redo
ctrl+left/right	Add entry to list or column to matrix
shift+ctrl+left/right	Add copy of current entry/column to to list/matrix
ctrl+up/down	Add row to matrix
shift+ctrl+up/down	Add copy of current row to matrix
ctrl+backspace	Delete current entry in list or column in matrix
ctrl+shift+backspace	Delete current row in matrix

Type...	...to get
norm	$\|\|\blue{[?]}\|\|$
text	$\text{\blue{[?]}}$
sym_name	$\backslash\texttt{\blue{[?]}}$
abs	$\left\|\blue{[?]}\right\|$
sqrt	$\sqrt{\blue{[?]}}$
paren	$\left(\blue{[?]}\right)$
floor	$\lfloor \blue{[?]} \rfloor$
factorial	$\blue{[?]}!$
exp	${\blue{[?]}}^{\blue{[?]}}$
sub	${\blue{[?]}}_{\blue{[?]}}$
frac	$\dfrac{\blue{[?]}}{\blue{[?]}}$
int	$\displaystyle\int{\blue{[?]}}d\blue{[?]}$
defi	$\displaystyle\int_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}d\blue{[?]}$
deriv	$\displaystyle\frac{d}{d\blue{[?]}}\blue{[?]}$
sum	$\displaystyle\sum_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
prod	$\displaystyle\prod_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
root	$\sqrt[\blue{[?]}]{\blue{[?]}}$
vec	$\left\langle \blue{[?]} \right\rangle$
mat	$\left(\begin{matrix} \blue{[?]} \end{matrix}\right)$
*	$\cdot$
infinity	$\infty$
arcsin	$\arcsin\left(\blue{[?]}\right)$
arccos	$\arccos\left(\blue{[?]}\right)$
arctan	$\arctan\left(\blue{[?]}\right)$
sin	$\sin\left(\blue{[?]}\right)$
cos	$\cos\left(\blue{[?]}\right)$
tan	$\tan\left(\blue{[?]}\right)$
sec	$\sec\left(\blue{[?]}\right)$
csc	$\csc\left(\blue{[?]}\right)$
cot	$\cot\left(\blue{[?]}\right)$
log	$\log\left(\blue{[?]}\right)$
ln	$\ln\left(\blue{[?]}\right)$
alpha	$\alpha$
beta	$\beta$
gamma	$\gamma$
delta	$\delta$
epsilon	$\epsilon$
zeta	$\zeta$
eta	$\eta$
theta	$\theta$
iota	$\iota$
kappa	$\kappa$
lambda	$\lambda$
mu	$\mu$
nu	$\nu$
xi	$\xi$
omicron	$\omicron$
pi	$\pi$
rho	$\rho$
sigma	$\sigma$
tau	$\tau$
upsilon	$\upsilon$
phi	$\phi$
chi	$\chi$
psi	$\psi$
omega	$\omega$
Gamma	$\Gamma$
Delta	$\Delta$
Theta	$\Theta$
Lambda	$\Lambda$
Xi	$\Xi$
Pi	$\Pi$
Sigma	$\Sigma$
Phi	$\Phi$
Psi	$\Psi$
Omega	$\Omega$

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