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Big Ideas Math Algebra 1, 2015

BI

Big Ideas Math Algebra 1, 2015 View details

Chapter Review

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Exercise 2 Page 282

By graphing the given equations, we can determine the number of solutions to the system. This will be the point at which the lines intersect. To do this, we will need the equations to be in slope-intercept form to help us identify the slope $m$ and $y -$ intercept $b .$

Write in Slope-Intercept Form

Let's rewrite each of the equations in the system in slope-intercept form, highlighting the $m$ and $b$ values.

Given Equation	Slope-Intercept Form	Slope $m$	$y -$ intercept $b$
$y = - 4 x + 3$	$y = - 4 x + 3$	$- 4$	$(0, 3)$
$4 x - 2 y = 6$	$y = 2 x + (- 3)$	$2$	$(0, - 3)$

Graphing the System

To graph these equations, we will start by plotting their $y -$ intercepts. Then, we will use the slope to determine another point that satisfies each equation, and connect the points with a line.

equations graph

We can see that the lines intersect at exactly one point.

intersection point

The point of intersection at

(1, - 1)

is the solution to the system.

Checking Our Answer

Checking Our Solution

In case we want to check our answers, we need to substitute the point of intersection into the original system of equations.

{y = - 4 x + 3 4 x - 2 y = 6 (I) (II)

$(I), (II):$ $x = 1$ , $y = - 1$

{- 1 = ? - 4 (1) + 3 4 (1) - 2 (- 1) = ? 6

$(I):$ $(- a) b = - a b$

{- 1 = ? - 4 + 3 4 (1) - 2 (- 1) = ? 6

MultNegNegOnePar

$(II):$ $- a (- b) = a \cdot b$

{- 1 = ? - 4 + 3 4 (1) + 2 = ? 6

$(II):$ $a \cdot 1 = a$

{- 1 = ? - 4 + 3 - 4 + 2 = ? 6

$(I), (II):$ Add and subtract terms

{- 1 = - 1 ✓ - 6 = - 6 ✓

Because our substitution resulted in two identities, we know that our answer is correct!

Showing Our Work

Writing Equations in Slope-Intercept Form

In case you want to see the steps we took to rewrite the equations in slope-intercept form, we have included these calculations.

{y = - 4 x + 3 4 x - 2 y = 6 (I) (II)

$(II):$ $LHS - 4 x = RHS - 4 x$

{y = - 4 x + 3 - 2 y = - 4 x + 6

$(II):$ $LHS / (- 2) = RHS / (- 2)$

{y = - 4 x + 3 y = 2 x - 3