Question

fpc19
systems of linear equations problem set
properties of linear systems

1. determine the number of solutions of each linear system.

a. ( x + 2y = 6 )
( x + y = -2 )
b. ( 3x + 5y = 9 )
( 6x + 10y = 18 )
c. ( 2x - 5y = 30 )
( 4x - 10y = 15 )
d. ( \frac{x}{2} + \frac{y}{3} = \frac{1}{2} )
( \frac{x}{2} + \frac{y}{3} = \frac{1}{4} )

Explanation:

To determine the number of solutions for each linear system, we analyze the slopes and intercepts (or use elimination/substitution) to see if the lines are parallel, coincident, or intersecting. Let's solve each part:

Part (a)

System:

$$
LATEXBLOCK0
$$

Step 1: Rewrite in slope-intercept form (\(y = mx + b\))

• First equation: \(2y = -x + 6 \implies y = -\frac{1}{2}x + 3\) (slope \(m_1 = -\frac{1}{2}\), intercept \(b_1 = 3\))
• Second equation: \(y = -x - 2\) (slope \(m_2 = -1\), intercept \(b_2 = -2\))

Step 2: Compare slopes and intercepts

Since \(m_1
eq m_2\) (slopes are different), the lines intersect at one point. Thus, the system has 1 solution.

Part (b)

System:

$$
LATEXBLOCK1
$$

Step 1: Simplify the second equation

Divide the second equation by 2: \(3x + 5y = 9\) (same as the first equation).

Step 2: Analyze the system

Both equations are identical (coincident lines). Thus, the system has infinitely many solutions.

Part (c)

System:

$$
LATEXBLOCK2
$$

Step 1: Rewrite in slope-intercept form

• First equation: \(-5y = -2x + 30 \implies y = \frac{2}{5}x - 6\) (slope \(m_1 = \frac{2}{5}\), intercept \(b_1 = -6\))
• Second equation: \(-10y = -4x + 15 \implies y = \frac{4}{10}x - \frac{15}{10} \implies y = \frac{2}{5}x - 1.5\) (slope \(m_2 = \frac{2}{5}\), intercept \(b_2 = -1.5\))

Step 2: Compare slopes and intercepts

Slopes are equal (\(m_1 = m_2\)), but intercepts are different (\(b_1
eq b_2\)). Thus, the lines are parallel and never intersect—the system has 0 solutions.

Part (d)

System:

$$
LATEXBLOCK3
$$

Step 1: Analyze the equations

Both equations have the form \(\frac{x}{2} + \frac{y}{3} = \text{constant}\), but the constants are different (\(\frac{1}{2}
eq \frac{1}{4}\)).

Step 2: Interpret the lines

These are parallel lines (same left-hand side, different right-hand side) and never intersect. Thus, the system has 0 solutions.

Final Answers

a. \(\boldsymbol{1}\) solution
b. \(\boldsymbol{\text{Infinitely many}}\) solutions
c. \(\boldsymbol{0}\) solutions
d. \(\boldsymbol{0}\) solutions

Answer:

To determine the number of solutions for each linear system, we analyze the slopes and intercepts (or use elimination/substitution) to see if the lines are parallel, coincident, or intersecting. Let's solve each part:

Part (a)

System:

$$
LATEXBLOCK0
$$

Step 1: Rewrite in slope-intercept form (\(y = mx + b\))

• First equation: \(2y = -x + 6 \implies y = -\frac{1}{2}x + 3\) (slope \(m_1 = -\frac{1}{2}\), intercept \(b_1 = 3\))
• Second equation: \(y = -x - 2\) (slope \(m_2 = -1\), intercept \(b_2 = -2\))

Step 2: Compare slopes and intercepts

Since \(m_1
eq m_2\) (slopes are different), the lines intersect at one point. Thus, the system has 1 solution.

Part (b)

System:

$$
LATEXBLOCK1
$$

Step 1: Simplify the second equation

Divide the second equation by 2: \(3x + 5y = 9\) (same as the first equation).

Step 2: Analyze the system

Both equations are identical (coincident lines). Thus, the system has infinitely many solutions.

Part (c)

System:

$$
LATEXBLOCK2
$$

Step 1: Rewrite in slope-intercept form

• First equation: \(-5y = -2x + 30 \implies y = \frac{2}{5}x - 6\) (slope \(m_1 = \frac{2}{5}\), intercept \(b_1 = -6\))
• Second equation: \(-10y = -4x + 15 \implies y = \frac{4}{10}x - \frac{15}{10} \implies y = \frac{2}{5}x - 1.5\) (slope \(m_2 = \frac{2}{5}\), intercept \(b_2 = -1.5\))

Step 2: Compare slopes and intercepts

Slopes are equal (\(m_1 = m_2\)), but intercepts are different (\(b_1
eq b_2\)). Thus, the lines are parallel and never intersect—the system has 0 solutions.

Part (d)

System:

$$
LATEXBLOCK3
$$

Step 1: Analyze the equations

Both equations have the form \(\frac{x}{2} + \frac{y}{3} = \text{constant}\), but the constants are different (\(\frac{1}{2}
eq \frac{1}{4}\)).

Step 2: Interpret the lines

These are parallel lines (same left-hand side, different right-hand side) and never intersect. Thus, the system has 0 solutions.

Final Answers

a. \(\boldsymbol{1}\) solution
b. \(\boldsymbol{\text{Infinitely many}}\) solutions
c. \(\boldsymbol{0}\) solutions
d. \(\boldsymbol{0}\) solutions