Solving Systems Of Equations And Finding Parallel Lines A Detailed Guide

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When dealing with a system of equations, our primary goal is to find the values of the variables that satisfy all equations simultaneously. In this case, we have the system:

3x−4y=−203x - 4y = -20

4x+3y=−64x + 3y = -6

To solve this system, we can use several methods, including substitution, elimination, and matrix methods. Here, we will use the elimination method, which is particularly effective when the coefficients of one of the variables are easy to manipulate to be opposites.

Elimination Method: A Step-by-Step Approach

The elimination method involves multiplying one or both equations by constants so that the coefficients of one variable are opposites. This allows us to eliminate that variable when we add the equations together.

  1. Identify a Variable to Eliminate: Look at the coefficients of x and y. Notice that the coefficients of y are -4 and 3. The least common multiple of 4 and 3 is 12, so we can make these coefficients opposites by multiplying the first equation by 3 and the second equation by 4.

  2. Multiply the Equations:

    • Multiply the first equation (3x−4y=−203x - 4y = -20) by 3:

    3(3x−4y)=3(−20)3(3x - 4y) = 3(-20)

    9x−12y=−609x - 12y = -60

    • Multiply the second equation (4x+3y=−64x + 3y = -6) by 4:

    4(4x+3y)=4(−6)4(4x + 3y) = 4(-6)

    16x+12y=−2416x + 12y = -24

  3. Add the Modified Equations: Now, add the two new equations:

    (9x−12y)+(16x+12y)=−60+(−24)(9x - 12y) + (16x + 12y) = -60 + (-24)

    9x−12y+16x+12y=−849x - 12y + 16x + 12y = -84

    25x=−8425x = -84

  4. Solve for x: Divide both sides by 25 to find the value of x:

    x=−8425x = \frac{-84}{25}

Solving for y: Substituting x Back into the Equation

Now that we have the value of x, we can substitute it back into one of the original equations to solve for y. Let’s use the first equation, 3x−4y=−203x - 4y = -20:

  1. Substitute x:

    3(−8425)−4y=−203(\frac{-84}{25}) - 4y = -20

    −25225−4y=−20\frac{-252}{25} - 4y = -20

  2. Isolate the y term: Add 25225\frac{252}{25} to both sides:

    −4y=−20+25225-4y = -20 + \frac{252}{25}

    −4y=−50025+25225-4y = \frac{-500}{25} + \frac{252}{25}

    −4y=−24825-4y = \frac{-248}{25}

  3. Solve for y: Divide both sides by -4:

    y=−24825÷−4y = \frac{-248}{25} \div -4

    y=−24825×−14y = \frac{-248}{25} \times \frac{-1}{4}

    y=248100y = \frac{248}{100}

    y=6225y = \frac{62}{25}

Thus, the solution to the system of equations is:

x=−8425,y=6225x = \frac{-84}{25}, \quad y = \frac{62}{25}

To find a line parallel to a given line, we first need to determine the slope of the given line. We can find the slope by rearranging one of the equations into slope-intercept form (y=mx+by = mx + b), where m represents the slope.

Let’s use the first equation, 3x−4y=−203x - 4y = -20, to find the slope:

  1. Isolate the y term:

    −4y=−3x−20-4y = -3x - 20

  2. Divide by -4:

    y=−3x−4+−20−4y = \frac{-3x}{-4} + \frac{-20}{-4}

    y=34x+5y = \frac{3}{4}x + 5

From this slope-intercept form, we can see that the slope of the given line is 34\frac{3}{4}.

Parallel Lines: Understanding the Concept

Parallel lines have the same slope. Therefore, any line parallel to the given line will also have a slope of 34\frac{3}{4}.

Now, we need to find the equation of a line that is parallel to the given line and passes through the point (-3, 1). We know the slope of the parallel line is 34\frac{3}{4}, and we have a point it passes through, so we can use the point-slope form of a linear equation:

y−y1=m(x−x1)y - y_1 = m(x - x_1)

where m is the slope, and (x1,y1)(x_1, y_1) is the point the line passes through.

Using the Point-Slope Form: A Step-by-Step Solution

  1. Plug in the values: Substitute the slope m=34m = \frac{3}{4} and the point (−3,1)(-3, 1) into the point-slope form:

    y−1=34(x−(−3))y - 1 = \frac{3}{4}(x - (-3))

    y−1=34(x+3)y - 1 = \frac{3}{4}(x + 3)

Analyzing the Answer Choices: Matching the Equation Form

Now, let's compare our equation with the given answer choices. The options are in the form:

A. $y - 1 = -\frac{3}{2}$ B. $y - 1 = -\frac{2}{3}$ C. $y - 1 = \frac{2}{3}$ D. $y - 1 = \frac{3}{2}$

Our equation is y−1=34(x+3)y - 1 = \frac{3}{4}(x + 3). None of the answer choices match this form directly. However, we made an error in our process. We need a line parallel to the original line, which means it must have the same slope when the original equations are solved for slope-intercept form. The slope we calculated was from the equation 3x−4y=−203x - 4y = -20, which we correctly found to be 34\frac{3}{4}. However, for a parallel line, the slopes should be the same, but there seems to be a mistake in the provided answer choices as none have the correct slope.

Let's re-examine the original equations to ensure we have the correct slope and then find the slope of a perpendicular line, which is likely what the question intended.

It appears the question may have intended to ask for a line perpendicular to the given line, rather than parallel. Let's correct our approach and find the equation of a line perpendicular to the original line and passing through the point (-3, 1).

Finding the Slope of a Perpendicular Line

The slope of a line perpendicular to a given line is the negative reciprocal of the slope of the given line. If the slope of the given line is 34\frac{3}{4}, the slope of the perpendicular line will be:

m⊥=−134=−43m_{\perp} = -\frac{1}{\frac{3}{4}} = -\frac{4}{3}

Using the Perpendicular Slope in the Point-Slope Form

Now, we use the point-slope form with the perpendicular slope m⊥=−43m_{\perp} = -\frac{4}{3} and the point (-3, 1):

y−1=−43(x−(−3))y - 1 = -\frac{4}{3}(x - (-3))

y−1=−43(x+3)y - 1 = -\frac{4}{3}(x + 3)

Let's look at the answer choices again:

A. $y - 1 = -\frac{3}{2}$ B. $y - 1 = -\frac{2}{3}$ C. $y - 1 = \frac{2}{3}$ D. $y - 1 = \frac{3}{2}$

We are looking for an equation in the form y−1=m(x+3)y - 1 = m(x + 3). None of these equations directly match the form we derived, y−1=−43(x+3)y - 1 = -\frac{4}{3}(x + 3). It seems there may be an issue with the provided answer choices or the question itself. However, let's analyze further by simplifying our equation and seeing if we can match a slope.

Further Analysis and Simplification

Expanding our equation, we get:

y−1=−43x−4y - 1 = -\frac{4}{3}x - 4

y=−43x−3y = -\frac{4}{3}x - 3

The slope of the perpendicular line is −43-\frac{4}{3}. None of the answer choices provide an equation with this slope. It is possible that the question intended to provide the slope directly without the (x+3)(x + 3) term, but even then, −43-\frac{4}{3} is not an option. The closest slope provided is −32-\frac{3}{2}, but this is not the correct negative reciprocal.

Based on our detailed analysis, it appears there may be an error in the provided answer choices or in the question's intention. If the question intended to ask for a line parallel to the given line, the slope should be 34\frac{3}{4}, which is not reflected in any of the options. If the question intended to ask for a line perpendicular to the given line, the slope should be −43-\frac{4}{3}, which is also not reflected in the options.

It is recommended to review the original question and answer choices for any potential errors or typos. If clarification is possible, it should be sought to ensure an accurate solution.

In summary, to solve a system of equations, methods like elimination can be used to find the values of variables that satisfy all equations. To find a line parallel to a given line, we use the same slope, and to find a line perpendicular, we use the negative reciprocal of the slope. Understanding these concepts is crucial for solving linear equations and related problems in mathematics.