# Chapter 6 - Systems of Equations and Inequalities - 6-1 Solving Systems by Graphing - Practice and Problem-Solving Exercises - Page 363: 10

After graphing the functions we find that the two intercept at $(2,4)$. The two functions are consistent and independent because they are two different functions that share one intersection point and have different slopes

#### Work Step by Step

#$1$ The standard notation for linear functions is $y=mx + b$, where m=slope of the function (rise over run) and b=the y-intercept of the function. #$2$ For Equation 1, $y= 2x$, the $m$ value is $2$, which means the slope is $2$, in other words this means for every increase of 1 in the positive x-direction the y-value will increase by $2$ in the positive y-direction. There is no $b$ value so the y-intercept is $(0,0)$. $y= 2x; slope=2 and y-intercept= (0,0)$ #$3$ Equation 2, $y= -2+8$, the $m$ value, or slope, is $-2$, which means for every increase of 1 in the positive x-direction the y-value will decrease by 2 There is $b$ value so the y-intercept is (0,8) $y= -2x+8; slope= -2 and y-intercept= (0,8)$ #$4$ When we graph the $2$ functions, $Equation 1$ corresponds with $f(x)=2x$ in the $blue color$ and we can see that it has a slope of $2$ and its y-intercept is at $(0,0)$. $Equation 2$ corresponds with $g(x)=8-2x$ in the $red color$ and it has a slope of $-2$ and its y-intercept is at $(0, 8)$. We can see in the graph that the two functions intersect at $(2, 4)$. This means that the two functions equal one another at this point. #$5$ We can double check our result by making the functions equal one another and solving for x. After doing this we find that the two functions equal one another when $x = 2$. By plugging this back into the functions we can check this again because when $x = 2$ both functions are $y = 4$. Solution- $2x = -2x + 8$ $+2x .. +2x$ $4x = 8$ $4x\div4=8\div4$ $x = 2$

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