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Sagot :
Certainly! To solve the given system of linear equations by graphing, follow these steps:
1. Graph the first equation [tex]\( y = 4x - 3 \)[/tex]:
- This is a straight line with a slope of 4 and a y-intercept of -3.
- To plot this line, you can use the intercept and the slope:
- The y-intercept is [tex]\( (0, -3) \)[/tex].
- From the y-intercept, use the slope (rise over run) to find another point. Since the slope is 4, it means for every 1 unit you move to the right (along the x-axis), you move 4 units up (along the y-axis). So, another point on the line is [tex]\( (1, 1) \)[/tex] because:
[tex]\[ y = 4(1) - 3 = 4 - 3 = 1 \][/tex]
- Plot the points [tex]\((0, -3)\)[/tex] and [tex]\((1, 1)\)[/tex] and draw the line through them.
2. Graph the second equation [tex]\( y = \frac{1}{2}x + 4 \)[/tex]:
- This is a straight line with a slope of [tex]\(\frac{1}{2}\)[/tex] and a y-intercept of 4.
- To plot this line, you can also use the intercept and the slope:
- The y-intercept is [tex]\( (0, 4) \)[/tex].
- From the y-intercept, use the slope to find another point. Since the slope is [tex]\(\frac{1}{2}\)[/tex], it means for every 2 units you move to the right (along the x-axis), you move 1 unit up (along the y-axis). Another point on the line is [tex]\( (2, 5) \)[/tex] because:
[tex]\[ y = \frac{1}{2}(2) + 4 = 1 + 4 = 5 \][/tex]
- Plot the points [tex]\((0, 4)\)[/tex] and [tex]\((2, 5)\)[/tex] and draw the line through them.
3. Find the intersection point:
- The point where these two lines intersect is the solution to the system of equations. By graphing both lines, you can see that they intersect at the point [tex]\((2, 5)\)[/tex].
4. Verification:
- To ensure the solution is correct, substitute [tex]\(x = 2\)[/tex] and [tex]\(y = 5\)[/tex] back into both original equations:
- For [tex]\( y = 4x - 3 \)[/tex]:
[tex]\[ 5 = 4(2) - 3 = 8 - 3 = 5 \][/tex]
- For [tex]\( y = \frac{1}{2}x + 4 \)[/tex]:
[tex]\[ 5 = \frac{1}{2}(2) + 4 = 1 + 4 = 5 \][/tex]
- Both equations are satisfied, confirming that the solution is correct.
Therefore, the solution to the system of equations is the ordered pair [tex]\((2, 5)\)[/tex].
1. Graph the first equation [tex]\( y = 4x - 3 \)[/tex]:
- This is a straight line with a slope of 4 and a y-intercept of -3.
- To plot this line, you can use the intercept and the slope:
- The y-intercept is [tex]\( (0, -3) \)[/tex].
- From the y-intercept, use the slope (rise over run) to find another point. Since the slope is 4, it means for every 1 unit you move to the right (along the x-axis), you move 4 units up (along the y-axis). So, another point on the line is [tex]\( (1, 1) \)[/tex] because:
[tex]\[ y = 4(1) - 3 = 4 - 3 = 1 \][/tex]
- Plot the points [tex]\((0, -3)\)[/tex] and [tex]\((1, 1)\)[/tex] and draw the line through them.
2. Graph the second equation [tex]\( y = \frac{1}{2}x + 4 \)[/tex]:
- This is a straight line with a slope of [tex]\(\frac{1}{2}\)[/tex] and a y-intercept of 4.
- To plot this line, you can also use the intercept and the slope:
- The y-intercept is [tex]\( (0, 4) \)[/tex].
- From the y-intercept, use the slope to find another point. Since the slope is [tex]\(\frac{1}{2}\)[/tex], it means for every 2 units you move to the right (along the x-axis), you move 1 unit up (along the y-axis). Another point on the line is [tex]\( (2, 5) \)[/tex] because:
[tex]\[ y = \frac{1}{2}(2) + 4 = 1 + 4 = 5 \][/tex]
- Plot the points [tex]\((0, 4)\)[/tex] and [tex]\((2, 5)\)[/tex] and draw the line through them.
3. Find the intersection point:
- The point where these two lines intersect is the solution to the system of equations. By graphing both lines, you can see that they intersect at the point [tex]\((2, 5)\)[/tex].
4. Verification:
- To ensure the solution is correct, substitute [tex]\(x = 2\)[/tex] and [tex]\(y = 5\)[/tex] back into both original equations:
- For [tex]\( y = 4x - 3 \)[/tex]:
[tex]\[ 5 = 4(2) - 3 = 8 - 3 = 5 \][/tex]
- For [tex]\( y = \frac{1}{2}x + 4 \)[/tex]:
[tex]\[ 5 = \frac{1}{2}(2) + 4 = 1 + 4 = 5 \][/tex]
- Both equations are satisfied, confirming that the solution is correct.
Therefore, the solution to the system of equations is the ordered pair [tex]\((2, 5)\)[/tex].
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