Answer :
To solve the system of linear equations
[tex]\[ \left\{\begin{array}{l} 0.2 x - 0.5 y = 10 \\ 0.5 x + 0.3 y = 15 \end{array}\right. \][/tex]
by graphing, follow these steps:
### Step 1: Rewrite each equation in slope-intercept form (y = mx + b)
For the first equation [tex]\( 0.2x - 0.5y = 10 \)[/tex]:
1. Isolate [tex]\( y \)[/tex]:
[tex]\[ 0.2x - 0.5y = 10 \implies -0.5y = -0.2x + 10 \implies y = \frac{-0.2x + 10}{-0.5} \][/tex]
2. Simplify:
[tex]\[ y = \frac{-0.2x + 10}{-0.5} \implies y = 0.4x - 20 \][/tex]
So the first equation in slope-intercept form is:
[tex]\[ y = 0.4x - 20 \][/tex]
For the second equation [tex]\( 0.5x + 0.3y = 15 \)[/tex]:
1. Isolate [tex]\( y \)[/tex]:
[tex]\[ 0.5x + 0.3y = 15 \implies 0.3y = -0.5x + 15 \implies y = \frac{-0.5x + 15}{0.3} \][/tex]
2. Simplify:
[tex]\[ y = \frac{-0.5x + 15}{0.3} \implies y = -\frac{5}{3}x + 50 \][/tex]
So the second equation in slope-intercept form is:
[tex]\[ y = -\frac{5}{3}x + 50 \][/tex]
### Step 2: Plot the graphs of these two lines
Graph of [tex]\( y = 0.4x - 20 \)[/tex]:
- Find the intercept points:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 0.4(0) - 20 = -20 \)[/tex]. So, the y-intercept is (0, -20).
- When [tex]\( y = 0 \)[/tex], [tex]\( 0 = 0.4x - 20 \implies x = 50 \)[/tex]. So, the x-intercept is (50, 0).
Graph of [tex]\( y = -\frac{5}{3}x + 50 \)[/tex]:
- Find the intercept points:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = -\frac{5}{3}(0) + 50 = 50 \)[/tex]. So, the y-intercept is (0, 50).
- When [tex]\( y = 0 \)[/tex], [tex]\( 0 = -\frac{5}{3}x + 50 \implies x = 30 \)[/tex]. So, the x-intercept is (30, 0).
### Step 3: Determine the point of intersection
By plotting these lines on a graph, you will see that they intersect at a specific point. Find this intersection point algebraically:
1. Set the two equations equal to each other to solve for [tex]\( x \)[/tex]:
[tex]\[ 0.4x - 20 = -\frac{5}{3}x + 50 \][/tex]
2. Combine like terms:
[tex]\[ 0.4x + \frac{5}{3}x = 70 \][/tex]
3. Simplify:
[tex]\[ \frac{1.2x + 5x}{3} = 70 \implies 1.2x + 5x = 210 \implies 6.2x = 210 \implies x = \frac{210}{6.2} \implies x \approx 33.87 \][/tex]
4. Substitute [tex]\( x \approx 33.87 \)[/tex] back into one of the original equations to find [tex]\( y \)[/tex]. Using [tex]\( y = 0.4x - 20 \)[/tex]:
[tex]\[ y = 0.4(33.87) - 20 \approx 13.548 - 20 \approx -6.45 \][/tex]
Thus, the intersection point and the solution to the system, rounded to the nearest hundredth, is:
[tex]\[ (x, y) \approx (33.87, -6.45) \][/tex]
### Step 4: Verify the solution
Verify the solution by substituting [tex]\( x \approx 33.87 \)[/tex] and [tex]\( y \approx -6.45 \)[/tex] into both of the original equations to confirm it satisfies both.
So, the correct solution rounded to the nearest hundredth is:
(33.87, -6.45)
Hence, the correct choice is:
[tex]\(\boxed{(33.87, -6.45)}\)[/tex]
[tex]\[ \left\{\begin{array}{l} 0.2 x - 0.5 y = 10 \\ 0.5 x + 0.3 y = 15 \end{array}\right. \][/tex]
by graphing, follow these steps:
### Step 1: Rewrite each equation in slope-intercept form (y = mx + b)
For the first equation [tex]\( 0.2x - 0.5y = 10 \)[/tex]:
1. Isolate [tex]\( y \)[/tex]:
[tex]\[ 0.2x - 0.5y = 10 \implies -0.5y = -0.2x + 10 \implies y = \frac{-0.2x + 10}{-0.5} \][/tex]
2. Simplify:
[tex]\[ y = \frac{-0.2x + 10}{-0.5} \implies y = 0.4x - 20 \][/tex]
So the first equation in slope-intercept form is:
[tex]\[ y = 0.4x - 20 \][/tex]
For the second equation [tex]\( 0.5x + 0.3y = 15 \)[/tex]:
1. Isolate [tex]\( y \)[/tex]:
[tex]\[ 0.5x + 0.3y = 15 \implies 0.3y = -0.5x + 15 \implies y = \frac{-0.5x + 15}{0.3} \][/tex]
2. Simplify:
[tex]\[ y = \frac{-0.5x + 15}{0.3} \implies y = -\frac{5}{3}x + 50 \][/tex]
So the second equation in slope-intercept form is:
[tex]\[ y = -\frac{5}{3}x + 50 \][/tex]
### Step 2: Plot the graphs of these two lines
Graph of [tex]\( y = 0.4x - 20 \)[/tex]:
- Find the intercept points:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 0.4(0) - 20 = -20 \)[/tex]. So, the y-intercept is (0, -20).
- When [tex]\( y = 0 \)[/tex], [tex]\( 0 = 0.4x - 20 \implies x = 50 \)[/tex]. So, the x-intercept is (50, 0).
Graph of [tex]\( y = -\frac{5}{3}x + 50 \)[/tex]:
- Find the intercept points:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = -\frac{5}{3}(0) + 50 = 50 \)[/tex]. So, the y-intercept is (0, 50).
- When [tex]\( y = 0 \)[/tex], [tex]\( 0 = -\frac{5}{3}x + 50 \implies x = 30 \)[/tex]. So, the x-intercept is (30, 0).
### Step 3: Determine the point of intersection
By plotting these lines on a graph, you will see that they intersect at a specific point. Find this intersection point algebraically:
1. Set the two equations equal to each other to solve for [tex]\( x \)[/tex]:
[tex]\[ 0.4x - 20 = -\frac{5}{3}x + 50 \][/tex]
2. Combine like terms:
[tex]\[ 0.4x + \frac{5}{3}x = 70 \][/tex]
3. Simplify:
[tex]\[ \frac{1.2x + 5x}{3} = 70 \implies 1.2x + 5x = 210 \implies 6.2x = 210 \implies x = \frac{210}{6.2} \implies x \approx 33.87 \][/tex]
4. Substitute [tex]\( x \approx 33.87 \)[/tex] back into one of the original equations to find [tex]\( y \)[/tex]. Using [tex]\( y = 0.4x - 20 \)[/tex]:
[tex]\[ y = 0.4(33.87) - 20 \approx 13.548 - 20 \approx -6.45 \][/tex]
Thus, the intersection point and the solution to the system, rounded to the nearest hundredth, is:
[tex]\[ (x, y) \approx (33.87, -6.45) \][/tex]
### Step 4: Verify the solution
Verify the solution by substituting [tex]\( x \approx 33.87 \)[/tex] and [tex]\( y \approx -6.45 \)[/tex] into both of the original equations to confirm it satisfies both.
So, the correct solution rounded to the nearest hundredth is:
(33.87, -6.45)
Hence, the correct choice is:
[tex]\(\boxed{(33.87, -6.45)}\)[/tex]
