GED Math : Algebra

Study concepts, example questions & explanations for GED Math

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Example Questions

Example Question #231 : Algebra

Line

Give the equation, in standard form, of the line on the above set of coordinate axes.

Possible Answers:

\displaystyle 5x+3y = 15

\displaystyle 5x-3y = 15

\displaystyle 3x-5y = 15

\displaystyle 3x+ 5y = 15

Correct answer:

\displaystyle 5x+3y = 15

Explanation:

The \displaystyle y-intercept of the line can be seen to be at the point five units above the origin, which is \displaystyle (0, 5). The \displaystyle x-intercept is at the point three units to the right of the origin, which is \displaystyle (3,0). From these intercepts, we can find slope \displaystyle m by setting \displaystyle b= 5,a = 3 in the formula

\displaystyle m = - \frac{b}{a}

The slope is

\displaystyle m = - \frac{5}{3}

Now, we can find the slope-intercept form of the line 

\displaystyle y = mx + b

By setting \displaystyle m = - \frac{5}{3}\displaystyle b= 5:

\displaystyle y = - \frac{5}{3}x +5

The standard form of a linear equation in two variables is

\displaystyle ax+by = c,

so in order to find the equation in this form, first, add \displaystyle - \frac{5}{3}x to both sides:

\displaystyle y + \frac{5}{3}x = - \frac{5}{3}x +5 + \frac{5}{3}x

\displaystyle \frac{5}{3}x+ y = 5

We can eliminate the fraction by multiplying both sides by 3:

\displaystyle 3 \left (\frac{5}{3}x+ y \right )= 3 \cdot 5

Distribute by multiplying:

\displaystyle 3 \cdot \frac{5}{3}x+3 \cdot y =15

\displaystyle 5x+3y = 15,

the correct equation.

 

 

Example Question #232 : Algebra

Write the given equation in standard form:  \displaystyle 3+y =2(2x-6)

Possible Answers:

\displaystyle -4x+y = 9

\displaystyle 4x+y=-9

\displaystyle 4x-2y =9

\displaystyle 4x+y = 15

\displaystyle -4x+y = -15

Correct answer:

\displaystyle -4x+y = -15

Explanation:

The equation in standard form is:  \displaystyle Ax+By =C

Simplify the right side by distribution.

\displaystyle 3+y =2(2x)-(2)(6)

\displaystyle y+3 = 4x-12

Subtract \displaystyle 4x on both sides.

\displaystyle y+3 -4x= 4x-12-4x

The equation becomes: 

\displaystyle -4x+y+3 = -12

Subtract 3 from both sides.

\displaystyle -4x+y+3-3 = -12-3

The answer is:  \displaystyle -4x+y = -15

Example Question #233 : Algebra

Given the point \displaystyle (1,3) with a slope of two, write the equation in standard form.  

Possible Answers:

\displaystyle 2x-3y =-1

\displaystyle -2x-2y = 1

\displaystyle -2x+y = 1

\displaystyle 2x+3y =-1

\displaystyle y=2x-1

Correct answer:

\displaystyle -2x+y = 1

Explanation:

We will first need to write the point-slope form to set up the equation.

\displaystyle y-y_1 = m(x-x_1)

Substitute the slope and point.

\displaystyle y-3= 2(x-1)

Simplify the right side.

\displaystyle y-3 = 2x-2

Add 3 on both sides.

\displaystyle y-3+3 = 2x-2+3

\displaystyle y=2x+1

Subtract \displaystyle 2x on both sides.

\displaystyle y-2x=2x+1-2x

The answer is:  \displaystyle -2x+y = 1

Example Question #12 : Standard Form

Find the equation in standard form:  \displaystyle y=4(x-4)

Possible Answers:

\displaystyle 2x+y = -4

\displaystyle -2x+y = -8

\displaystyle -4x+y = -4

\displaystyle -4x+y = -16

\displaystyle 2x+y = -16

Correct answer:

\displaystyle -4x+y = -16

Explanation:

Distribute the right side.

\displaystyle y=4x-16

Subtract \displaystyle 4x on both sides.

\displaystyle y-4x=4x-16-4x

The answer is:  \displaystyle -4x+y = -16

Example Question #234 : Algebra

Rewrite the equation in standard form:  \displaystyle \frac{1}{2}x = 2x-2y+3

Possible Answers:

\displaystyle 4x-4y =-6

\displaystyle y=\frac{3}{4}x +\frac{3}{2}

\displaystyle \frac{3}{4}x +\frac{3}{2}y =1

\displaystyle 4x+4y =-6

Correct answer:

\displaystyle 4x-4y =-6

Explanation:

The standard form of a linear equation is:  \displaystyle Ax+By=C

Multiply by two on both sides to eliminate the fraction.

\displaystyle (\frac{1}{2}x)(2) =2( 2x-2y+3)

\displaystyle x= 4x-4y+6

Subtract \displaystyle x on both sides.

\displaystyle x-x= 4x-4y+6-x

\displaystyle 0 = 4x-4y +6

Subtract 6 from both sides.

The answer is:  \displaystyle 4x-4y =-6

Example Question #11 : Standard Form

Given the slope is 3, and the y-intercept is 6, write the equation of the line in standard form.

Possible Answers:

\displaystyle x-3y=6

\displaystyle 3x+y= 6

\displaystyle x+3y=6

\displaystyle 3x-y= 6

\displaystyle -3x+y= 6

Correct answer:

\displaystyle -3x+y= 6

Explanation:

The standard form of a line is:  \displaystyle Ax+By =C

First, we can write the equation in slope-intercept form:  \displaystyle y=mx+b

\displaystyle y=3x+6

Subtract \displaystyle 3x on both sides.

\displaystyle y-3x=3x+6-3x

The answer is:  \displaystyle -3x+y= 6

Example Question #235 : Algebra

Given the slope of a line is 7, and a known point is (2,5), what is the equation of the line in standard form?

Possible Answers:

\displaystyle -7x+y =-19

\displaystyle -7x+y =-6

\displaystyle -7x+2y =-5

\displaystyle -7x+y =-9

\displaystyle 7x-3y =-5

Correct answer:

\displaystyle -7x+y =-9

Explanation:

The standard form of a line is:  \displaystyle Ax+By =C

We can use the point-slope form of a line since we are only given the slope and a point.

\displaystyle y-y_1 = m( x-x_1)

Substitute the slope and the point.

\displaystyle y-5 = 7( x-2)

Simplify this equation.

\displaystyle y-5 = 7x-14

Add \displaystyle 5 on both sides.

\displaystyle y-5+5= 7x-14+5

\displaystyle y=7x-9

Subtract \displaystyle 7x from both sides.

\displaystyle y- 7x=7x-9- 7x

Simplify both sides.

\displaystyle -7x+y =-9

The answer is:  \displaystyle -7x+y =-9

Example Question #61 : Linear Algebra

Rewrite the equation in standard form:  \displaystyle y= 4(x-2)

Possible Answers:

\displaystyle -4x+y = -2

\displaystyle -4x+y =2

\displaystyle -8x+4y = -1

\displaystyle -x+4y = -8

\displaystyle -4x+y = -8

Correct answer:

\displaystyle -4x+y = -8

Explanation:

Distribute the four through both terms of the binomial.

\displaystyle y=4x-8

Subtract \displaystyle 4x on both sides of the equation.

\displaystyle y-4x=4x-8-4x

Simplify both sides.

\displaystyle -4x+y = -8

The answer is:  \displaystyle -4x+y = -8

Example Question #23 : Standard Form

Which of the following is an equation, in standard form, of the line of the coordinate plane with intercepts \displaystyle (-3, 0 ) and \displaystyle (0, 8 ) ?

Possible Answers:

\displaystyle 8 x + 3 y =24

\displaystyle 8 x - 3 y =-24

\displaystyle 8 x + 3 y =-24

\displaystyle 8 x - 3 y =24

Correct answer:

\displaystyle 8 x - 3 y =-24

Explanation:

First, find the slope-intercept form of the equation. This is 

\displaystyle y = mx+b,

where \displaystyle m is the slope and \displaystyle (0,b) is the \displaystyle y-intercept of the line. Since \displaystyle (0, 8 ) is this intercept, \displaystyle b = 8. Also, the slope of a line with intercepts \displaystyle (a,0 ) and \displaystyle (0,b) is \displaystyle m = - \frac{b}{a}, so, setting \displaystyle a = -3, b = 8,

\displaystyle m = - \frac{8}{-3} = \frac{8}{3}.

The slope-intercept form is 

\displaystyle y = \frac{8}{3} x + 8

The standard form of the equation is 

\displaystyle Ax+By = C,

where, by custom, \displaystyle A\displaystyle B, and \displaystyle C are relatively prime integers, and \displaystyle A > 0. To accomplish this:

Switch the expressions:

\displaystyle \frac{8}{3} x + 8 = y

Add \displaystyle -y-8 to both sides:

\displaystyle \frac{8}{3} x + 8+ (-y-8 ) = y + (-y-8 )

\displaystyle \frac{8}{3} x - y =-8

Multiply both sides by 3 to eliminate the denominator and make the coefficients integers with GCF 1:

\displaystyle 3 \cdot \left (\frac{8}{3} x - y \right ) =3 \cdot( -8 )

Distribute on the left:

\displaystyle 3 \cdot \frac{8}{3} x - 3 \cdot y =3 \cdot( -8 )

\displaystyle 8 x - 3 y =-24

This is the correct equation.

Example Question #236 : Algebra

Rewrite the equation

\displaystyle y = \frac{2}{7}x- \frac{1}{14}

in standard form so that the coefficients are integers, the coefficient of \displaystyle x is positive, and the three integers are relatively prime.

Possible Answers:

\displaystyle 4x-14y =1

\displaystyle 14x+4y = 1

\displaystyle 4x+ 14y = 1

\displaystyle 14x- 4y = 1

Correct answer:

\displaystyle 4x-14y =1

Explanation:

The standard form of the equation of a line is

\displaystyle ax+ by = c.

To rewrite the equation

\displaystyle y = \frac{2}{7}x- \frac{1}{14}

in this form so that \displaystyle x has a positive coefficient, first, switch the places of the expressions:

\displaystyle \frac{2}{7}x- \frac{1}{14} = y

Get the \displaystyle y term on the left and the constant on the right by adding \displaystyle -y+ \frac{1}{14} to both sides:

\displaystyle \frac{2}{7}x- \frac{1}{14} + \left ( -y+ \frac{1}{14} \right ) = y+ \left ( -y+ \frac{1}{14} \right )

\displaystyle \frac{2}{7}x- y = \frac{1}{14}

To eliminate fractions and ensure that the coefficients are  relatively prime, multiply both sides by lowest common denominator 14:

\displaystyle 14 \cdot \left (\frac{2}{7}x- y \right ) = 14 \cdot \frac{1}{14}

\displaystyle 14 \left (\frac{2}{7}x- y \right ) = \frac{14}{1} \cdot \frac{1}{14}

\displaystyle 14 \cdot \left (\frac{2}{7}x- y \right ) =1

Multiply 14 by both expressions in the parentheses:

\displaystyle 14 \cdot \frac{2}{7}x-14 \cdot y =1

\displaystyle \frac{14}{1} \cdot \frac{2}{7}x-14y =1

Cross-canceling:

\displaystyle \frac{2}{1} \cdot \frac{2}{1}x-14y =1

\displaystyle 4x-14y =1,

the correct choice.

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