Asked by Makayla
Solve the linear system of equations using addition. Graph the equations to verify your solution.
{-4x = y + 3
{y = -4x - 3
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Answered by
GPT-5 mini
AI
First rewrite both equations in a common form.
Equation (1): -4x = y + 3 => -4x - y - 3 = 0
Equation (2): y = -4x - 3 => 4x + y + 3 = 0
Add the two equations:
(-4x - y - 3) + (4x + y + 3) = 0 + 0 => 0 = 0.
Since adding them gives the identity 0 = 0, the two equations are dependent (one is a multiple of the other). In fact both describe the same line: y = -4x - 3.
Solution: infinitely many solutions — every point on the line y = -4x - 3. In set form:
{(x,y) | y = -4x - 3} or {(t, -4t - 3) | t ∈ R}.
Graph check: the line has slope −4 and y‑intercept (0, −3). Example points on the line: (0, −3), (1, −7), (−1, 1). Both equations produce the same line, so the graphs coincide, confirming infinitely many solutions.
Equation (1): -4x = y + 3 => -4x - y - 3 = 0
Equation (2): y = -4x - 3 => 4x + y + 3 = 0
Add the two equations:
(-4x - y - 3) + (4x + y + 3) = 0 + 0 => 0 = 0.
Since adding them gives the identity 0 = 0, the two equations are dependent (one is a multiple of the other). In fact both describe the same line: y = -4x - 3.
Solution: infinitely many solutions — every point on the line y = -4x - 3. In set form:
{(x,y) | y = -4x - 3} or {(t, -4t - 3) | t ∈ R}.
Graph check: the line has slope −4 and y‑intercept (0, −3). Example points on the line: (0, −3), (1, −7), (−1, 1). Both equations produce the same line, so the graphs coincide, confirming infinitely many solutions.
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